${}$

最近少し話題の, 手書きの数式を$\LaTeX$に変換してくれるサイトさん　(https://webdemo.myscript.com/views/math/index.html)　とペンタブを使って, 手軽に数式が打てそうだと思ったので, お気に入りの級数のまとめを作ってみました.
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<div style="padding: 0.2em 0.5em;
    margin: 2em 0;
    background: -webkit-repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    background: repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    box-shadow: 0px 0px 0px 5px #e9f4ff;
    border: dashed 2px white;">$$\TextCenter\sum ^{\infty }_{n=0}\dfrac{n!^{2}}{\left( 2n\right) !}=\dfrac{4}{3}+\dfrac{2\pi }{9\sqrt{3}}$$</div>

<div style="padding: 0.2em 0.5em;
    margin: 2em 0;
    background: -webkit-repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    background: repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    box-shadow: 0px 0px 0px 5px #e9f4ff;
    border: dashed 2px white;">$$\TextCenter\sum ^{\infty }_{n=0}\dfrac{2^{n}n!^{2}}{\left( 2n\right) !}=2+\dfrac{\pi }{2}$$</div>

<div style="padding: 0.2em 0.5em;
    margin: 2em 0;
    background: -webkit-repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    background: repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    box-shadow: 0px 0px 0px 5px #e9f4ff;
    border: dashed 2px white;">$$\TextCenter\sum ^{\infty }_{n=0}\dfrac{\left( -1\right) ^{n}n!^{2}}{\left( 2n\right) !}=\dfrac{4}{5}\left( 1-\dfrac{\log \varphi }{\sqrt{5}}\right)$$</div>
${}$
<div style="padding: 0.2em 0.5em;
    margin: 2em 0;
    background: -webkit-repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    background: repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    box-shadow: 0px 0px 0px 5px #e9f4ff;
    border: dashed 2px white;">$$\TextCenter\sum ^{\infty }_{n=0}\dfrac{\left( 5n+3\right) n!\left( 2n\right) !}{2^{n}\left( 3n+2\right) !}=\dfrac{\pi }{2}$$</div>

<div style="padding: 0.2em 0.5em;
    margin: 2em 0;
    background: -webkit-repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    background: repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    box-shadow: 0px 0px 0px 5px #e9f4ff;
    border: dashed 2px white;">$$\TextCenter\sum ^{\infty }_{n=0}\dfrac{\left( 5n+4\right) n!\left( 2n\right) !}{2^{n}\left( 3n+2\right) !}=3\log 2$$</div>
${}$
<div style="padding: 0.2em 0.5em;
    margin: 2em 0;
    background: -webkit-repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    background: repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    box-shadow: 0px 0px 0px 5px #e9f4ff;
    border: dashed 2px white;">$$\TextCenter\sum ^{\infty }_{n=0}\dfrac{n!\left( 2n\right) !}{2^{n}\left( 3n\right) !}=\dfrac{3}{5^{3}}\left( \dfrac{11}{6}\pi -2\log 2+45\right)$$</div>
${}$
<div style="padding: 0.2em 0.5em;
    margin: 2em 0;
    background: -webkit-repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    background: repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    box-shadow: 0px 0px 0px 5px #e9f4ff;
    border: dashed 2px white;">$$\TextCenter\sum ^{\infty }_{n=1}\dfrac{1}{2n-1}\left( \dfrac{\left( 2n\right) !}{2^{2n}n!^{2}}\right) =1$$</div>

<div style="padding: 0.2em 0.5em;
    margin: 2em 0;
    background: -webkit-repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    background: repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    box-shadow: 0px 0px 0px 5px #e9f4ff;
    border: dashed 2px white;">$$\TextCenter\sum ^{\infty }_{n=1}\dfrac{1}{2n-1}\left( \dfrac{\left( 2n\right) !}{2^{2n}n!^{2}}\right) ^{2}=1-\dfrac{2}{\pi }$$</div>
${}$
<div style="padding: 0.2em 0.5em;
    margin: 2em 0;
    background: -webkit-repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    background: repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    box-shadow: 0px 0px 0px 5px #e9f4ff;
    border: dashed 2px white;">$$\TextCenter\sum ^{\infty }_{n=1}\dfrac{H_{n}H_{n+1}}{n\left( n+1\right) }=\zeta(2) +2\zeta(3)
$$</div>

<div style="padding: 0.2em 0.5em;
    margin: 2em 0;
    background: -webkit-repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    background: repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    box-shadow: 0px 0px 0px 5px #e9f4ff;
    border: dashed 2px white;">$$\TextCenter\sum ^{\infty }_{n=1}\dfrac{H_{n}H_{n+1}H_{n+2}}{n\left( n+1\right) \left( n+2\right) }=-\dfrac{1}{2}\zeta(2) +\dfrac{5}{4}\zeta(3) +\dfrac{5}{8}\zeta(4)$$</div>
${}$
<div style="padding: 0.2em 0.5em;
    margin: 2em 0;
    background: -webkit-repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    background: repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    box-shadow: 0px 0px 0px 5px #e9f4ff;
    border: dashed 2px white;">$$\TextCenter\sum ^{\infty }_{n=1}\dfrac{\left( 2n\right) !}{2^{2n}n!^{2}}\dfrac{H_{n}}{n+1}=4\log 2$$</div>

<div style="padding: 0.2em 0.5em;
    margin: 2em 0;
    background: -webkit-repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    background: repeating-linear-gradient(-45deg, #f0f8ff, #f0f8ff 3px,#e9f4ff 3px, #e9f4ff 7px);
    box-shadow: 0px 0px 0px 5px #e9f4ff;
    border: dashed 2px white;">$$\TextCenter\sum ^{\infty }_{n=1}\dfrac{\left( 2n\right) !}{2^{2n}n!^{2}}\dfrac{H_{n}^{2}}{n+1}=\pi ^{2}+4\log ^{2}2$$</div>
${}$










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