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arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_01.html" aria-label="环境光渲染(一)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_02.html" aria-label="环境光渲染(二)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">编程语言 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">JavaScript <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/helloworld.html" aria-label="环境搭建"><!--[--><!--[--><!--]--><!--]-->环境搭建<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/grammar.html" aria-label="基本语法"><!--[--><!--[--><!--]--><!--]-->基本语法<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/function.html" aria-label="函数"><!--[--><!--[--><!--]--><!--]-->函数<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/object_and_class.html" aria-label="对象和类"><!--[--><!--[--><!--]--><!--]-->对象和类<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul><!--[--><!--]--></aside><!--]--><!--[--><main class="vp-page"><!--[--><!--]--><div vp-content><!--[--><!--]--><div><p>斐波那契数列作为最有名的数列之一广为人之，它的顺序就是 0,1,1,2,3,5,8,13,… 每个数字都是前两个数字的和，这个数列有很多有意思的特征，比和数字“89”的关系。方法很简单，把数列排列成一列，然后每个数都依次右移，最后加在一起形成一个小数，这个数恰好就是1/89。</p><div class="language-text" data-highlighter="prismjs" data-ext="text" data-title="text"><pre><code><span class="line"> 0.0</span>
|
||
<span class="line"> 0.01</span>
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||
<span class="line"> 0.001</span>
|
||
<span class="line"> 0.0002</span>
|
||
<span class="line"> 0.00003</span>
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||
<span class="line"> 0.000005</span>
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||
<span class="line"> 0.0000008</span>
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||
<span class="line"> 0.00000013</span>
|
||
<span class="line"> 0.000000021</span>
|
||
<span class="line"> 0.0000000034</span>
|
||
<span class="line"> ...</span>
|
||
<span class="line"> ----------------</span>
|
||
<span class="line"> 0.01123595505618... = 1/89</span>
|
||
<span class="line"></span></code></pre></div><p>这个是很容易证明的，我们知道Fibonacci数列的一般形式如下</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:21.198em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:21.198em;" side="right"><mjx-table style="width:auto;min-width:17.042em;margin:0 2.078em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c 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data-mjx-texclass="CLOSE">)</mo></mrow><mi>n</mi></msup><mrow data-mjx-texclass="ORD"><mo>–</mo></mrow><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mfrac><mrow><mn>1</mn><mo>−</mo><msqrt><mn>5</mn></msqrt></mrow><mn>2</mn></mfrac><mo data-mjx-texclass="CLOSE">)</mo></mrow><mi>n</mi></msup><mo data-mjx-texclass="CLOSE">]</mo></mrow></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>设<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msqrt space="3"><mjx-sqrt><mjx-surd><mjx-mo class="mjx-n"><mjx-c class="mjx-c221A"></mjx-c></mjx-mo></mjx-surd><mjx-box style="padding-top:0.163em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-box></mjx-sqrt></mjx-msqrt><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D6FD TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msqrt space="3"><mjx-sqrt><mjx-surd><mjx-mo class="mjx-n"><mjx-c class="mjx-c221A"></mjx-c></mjx-mo></mjx-surd><mjx-box style="padding-top:0.163em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-box></mjx-sqrt></mjx-msqrt><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msqrt><mn>5</mn></msqrt><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn><mo>,</mo><mi>β</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msqrt><mn>5</mn></msqrt><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>2</mn></math></mjx-assistive-mml></mjx-container>，那么</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:12.54em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:12.54em;" side="right"><mjx-table style="width:auto;min-width:8.384em;margin:0 2.078em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msqrt><mjx-sqrt><mjx-surd><mjx-mo class="mjx-n"><mjx-c class="mjx-c221A"></mjx-c></mjx-mo></mjx-surd><mjx-box style="padding-top:0.163em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-box></mjx-sqrt></mjx-msqrt></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msup space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FD TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:12.54em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.362em;"><mjx-mtd id="mjx-eqn:2"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.362em;vertical-align:-1.02em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(2)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><msqrt><mn>5</mn></msqrt></mfrac></mstyle><mo stretchy="false">(</mo><msup><mi>α</mi><mi>n</mi></msup><mo>−</mo><msup><mi>β</mi><mi>n</mi></msup><mo stretchy="false">)</mo></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>对于最开始提到的那个小数，可以描述为下面的方式</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:23.102em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:23.102em;" side="right"><mjx-table style="width:auto;min-width:18.946em;margin:0 2.078em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D446 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msup><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.393em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mstyle space="3"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msup><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.393em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mstyle space="3"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msup><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.393em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22EF"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-munderover space="2"><mjx-over style="padding-bottom:0.192em;padding-left:0.368em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c221E"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.058em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder></mjx-box></mjx-munderover><mjx-mstyle space="2"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msup><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.393em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:23.102em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.808em;"><mjx-mtd id="mjx-eqn:3"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.808em;vertical-align:-1.246em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(3)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mi>S</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msub><mi>a</mi><mn>0</mn></msub><msup><mn>10</mn><mn>1</mn></msup></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msub><mi>a</mi><mn>1</mn></msub><msup><mn>10</mn><mn>2</mn></msup></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><msub><mi>a</mi><mn>2</mn></msub><msup><mn>10</mn><mn>3</mn></msup></mfrac></mstyle><mo>+</mo><mo>⋯</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mn>10</mn></mfrac></mstyle><munderover><mo data-mjx-texclass="OP" movablelimits="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover><mstyle displaystyle="true" scriptlevel="0"><mfrac><msub><mi>a</mi><mi>n</mi></msub><msup><mn>10</mn><mi>n</mi></msup></mfrac></mstyle></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>把公式2带入公式3，可以得到</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:32.167em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:32.167em;" side="right"><mjx-table style="width:auto;min-width:28.011em;margin:0 2.078em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D446 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-munderover space="2"><mjx-over style="padding-bottom:0.192em;padding-left:0.368em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c221E"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.058em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder></mjx-box></mjx-munderover><mjx-mstyle space="2"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msqrt><mjx-sqrt><mjx-surd><mjx-mo class="mjx-n"><mjx-c class="mjx-c221A"></mjx-c></mjx-mo></mjx-surd><mjx-box style="padding-top:0.163em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-box></mjx-sqrt></mjx-msqrt></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msup><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.393em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msup space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FD TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-msqrt><mjx-sqrt><mjx-surd><mjx-mo class="mjx-n"><mjx-c class="mjx-c221A"></mjx-c></mjx-mo></mjx-surd><mjx-box style="padding-top:0.163em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-box></mjx-sqrt></mjx-msqrt></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mrow><mjx-mo class="mjx-s4"><mjx-c class="mjx-c5B TEX-S4"></mjx-c></mjx-mo><mjx-munderover><mjx-over style="padding-bottom:0.192em;padding-left:0.368em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c221E"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.058em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder></mjx-box></mjx-munderover><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-munderover space="3"><mjx-over style="padding-bottom:0.192em;padding-left:0.368em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c221E"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.058em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder></mjx-box></mjx-munderover><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FD TEX-I"></mjx-c></mjx-mi></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-s4"><mjx-c class="mjx-c5D TEX-S4"></mjx-c></mjx-mo></mjx-mrow></mjx-TeXAtom></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:32.167em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.999em;"><mjx-mtd id="mjx-eqn:4"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.999em;vertical-align:-1.25em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(4)</mtext></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mi>S</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mn>10</mn></mfrac></mstyle><munderover><mo data-mjx-texclass="OP" movablelimits="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><msqrt><mn>5</mn></msqrt></mfrac></mstyle><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><msup><mn>10</mn><mi>n</mi></msup></mfrac></mstyle><mo stretchy="false">(</mo><msup><mi>α</mi><mi>n</mi></msup><mo>−</mo><msup><mi>β</mi><mi>n</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mn>10</mn><msqrt><mn>5</mn></msqrt></mrow></mfrac></mstyle><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><munderover><mo data-mjx-texclass="OP" movablelimits="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover><mo stretchy="false">(</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>α</mi><mn>10</mn></mfrac></mstyle><msup><mo stretchy="false">)</mo><mi>n</mi></msup><mo>−</mo><munderover><mo data-mjx-texclass="OP" movablelimits="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover><mo stretchy="false">(</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mi>β</mi><mn>10</mn></mfrac></mstyle><msup><mo stretchy="false">)</mo><mi>n</mi></msup><mo data-mjx-texclass="CLOSE">]</mo></mrow></mrow></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container><p>由于<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math></mjx-assistive-mml></mjx-container>和<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FD TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math></mjx-assistive-mml></mjx-container>都小于10，根据等比数列的求和计算方法，可以得到</p><mjx-container class="MathJax" jax="CHTML" display="true" width="full" style="min-width:32.561em;position:relative;"><mjx-math width="full" display="true" class="MJX-TEX" aria-hidden="true"><mjx-mtable width="full" style="min-width:32.561em;" side="right"><mjx-table style="width:auto;min-width:28.405em;margin:0 2.078em;"><mjx-itable width="full"><mjx-mlabeledtr style="height:NaNem;"><mjx-mtd><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D446 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-msqrt><mjx-sqrt><mjx-surd><mjx-mo class="mjx-n"><mjx-c class="mjx-c221A"></mjx-c></mjx-mo></mjx-surd><mjx-box style="padding-top:0.163em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-box></mjx-sqrt></mjx-msqrt></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mrow><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5B TEX-S3"></mjx-c></mjx-mo><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2013"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D6FD TEX-I"></mjx-c></mjx-mi><mjx-TeXAtom texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2F"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5D TEX-S3"></mjx-c></mjx-mo></mjx-mrow><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msqrt><mjx-sqrt><mjx-surd><mjx-mo class="mjx-n"><mjx-c class="mjx-c221A"></mjx-c></mjx-mo></mjx-surd><mjx-box style="padding-top:0.163em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c35"></mjx-c></mjx-mn></mjx-box></mjx-sqrt></mjx-msqrt></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mstyle><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D6FD TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D6FC TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D6FD TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mstyle space="4"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c38"></mjx-c><mjx-c class="mjx-c39"></mjx-c></mjx-mn></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mstyle><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mlabeledtr></mjx-itable></mjx-table><mjx-labels style="width:32.561em;"><mjx-itable align="right" style="right:0;"><mjx-mtr style="height:2.47em;"><mjx-mtd id="mjx-eqn:5"><mjx-mtext class="mjx-n"><mjx-c class="mjx-c28"></mjx-c><mjx-c class="mjx-c35"></mjx-c><mjx-c class="mjx-c29"></mjx-c></mjx-mtext><mjx-tstrut style="height:2.47em;vertical-align:-1.02em;"></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-labels></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true"><mlabeledtr><mtd><mtext>(5)</mtext></mtd><mtd><mi>S</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mn>10</mn><msqrt><mn>5</mn></msqrt></mrow></mfrac></mstyle><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><mi>α</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>10</mn></mrow></mfrac></mstyle><mrow data-mjx-texclass="ORD"><mo>–</mo></mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><mi>β</mi><mrow data-mjx-texclass="ORD"><mo>/</mo></mrow><mn>10</mn></mrow></mfrac></mstyle><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><msqrt><mn>5</mn></msqrt></mfrac></mstyle><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mi>α</mi><mo>−</mo><mi>β</mi></mrow><mrow><mo stretchy="false">(</mo><mn>10</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>10</mn><mo>−</mo><mi>β</mi><mo stretchy="false">)</mo></mrow></mfrac></mstyle><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mn>89</mn></mfrac></mstyle></mtd></mlabeledtr></mtable></math></mjx-assistive-mml></mjx-container></div><!--[--><!--]--></div><footer class="vp-page-meta"><!----><div class="vp-meta-item git-info"><div class="vp-meta-item last-updated"><span class="meta-item-label">Last Updated: </span><!----></div><div class="vp-meta-item contributors"><span class="meta-item-label">Contributors: </span><span class="meta-item-info"><!--[--><!--[--><span class="contributor" title="email: thejinchao@gmail.com">thejinchao</span><!----><!--]--><!--]--></span></div></div></footer><nav class="vp-page-nav" aria-label="page navigation"><a class="route-link auto-link prev" href="/blog/2025/02/Ellipse.html" aria-label="一道数学趣题"><!--[--><div class="hint"><span class="arrow left"></span> Prev</div><div class="link"><span>一道数学趣题</span></div><!--]--></a><!----></nav><!--[--><!--]--></main><!--]--></div><!--[--><!----><!--]--><!--]--></div>
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