\(\href{https://showanojoe.com/template-math/double-integral/#区分求積法}{\color{teal}\textsf{区分求積法}}\)
\(\overset{{\color{red}{\small\displaystyle\int_{0}^{1}}}{\Large f}{\small({\color{blue}{x}})}\,{\color{yellow}{dx}}}{\scriptsize\textsf{定積分の定義式}}\)\(\hspace{10pt}\Leftarrow\hspace{10pt}\) \(\overset{{\it{lim}}_{\overset{\Large{~^~}}{\hspace{-25pt}{\scriptsize{n\scriptsize{\rightarrow}\scriptsize \infty}}}}\!\!{\color{yellow}{\displaystyle\frac{1}{n}}}{\color{red}{\displaystyle\sum_{i=1}^{n}}}\small {\large f}\left({\color{blue}{\scriptsize\displaystyle\frac{i}{n}}}\right)}{\scriptsize\textsf{区分求積法$($定積分の定義式の構造$)$}}\)
\({\small\displaystyle\frac{1}{n}}{\displaystyle\sum_{i={\tiny\displaystyle\frac{1}{2}}n}^{{\tiny\displaystyle\frac{7}{8}}n}}\small cos^{2}\left({\scriptsize\displaystyle\frac{i\pi}{n}}\right)\)\({\small\;\Rightarrow\left\{\begin{array}{l}\textsf{級数を表すために、初期値$\scriptsize\displaystyle\frac{n}{2}$ と 終了値$\scriptsize\displaystyle\frac{7}{8}n$ の分母を通分$($同じ数に$)$する。} \\ \textsf{とりあえず分母を$400$にすると、}\end{array}\right\}}\)
\({\small\hspace{10pt}\Rightarrow\hspace{10pt}}{\small\displaystyle\frac{1}{n}}{\displaystyle\sum_{i={\tiny\displaystyle\frac{200}{400}}n}^{{\tiny\displaystyle\frac{350}{400}}n}}\small cos^{2}\left({\scriptsize\displaystyle\frac{i\pi}{n}}\right)\)\({\small\;=\;}{\small\displaystyle\frac{1}{n}}{\huge\lbrace}{\small cos^{2}\left({\scriptsize\displaystyle\frac{{\tiny\displaystyle\frac{200n\pi}{400}}}{n}}\right)}{\small\;+\;}{\small cos^{2}\left({\scriptsize\displaystyle\frac{{\tiny\displaystyle\frac{201n\pi}{400}}}{n}}\right)}{\small\;+\;}{\small cos^{2}\left({\scriptsize\displaystyle\frac{{\tiny\displaystyle\frac{202n\pi}{400}}}{n}}\right)}{\small\;+\;}\)\(\cdots\cdots{\small\;+\;}{\small cos^{2}\left({\scriptsize\displaystyle\frac{{\tiny\displaystyle\frac{349n\pi}{400}}}{n}}\right)}{\small\;+\;}{\small cos^{2}\left({\scriptsize\displaystyle\frac{{\tiny\displaystyle\frac{350n\pi}{400}}}{n}}\right)}\)\({\huge\rbrace}\)
\({\small\displaystyle\frac{1}{n}}{\displaystyle\sum_{i={\tiny\displaystyle\frac{1}{2}}n}^{{\tiny\displaystyle\frac{7}{8}}n}}\small cos^{2}\left({\scriptsize\displaystyle\frac{i\pi}{n}}\right)\)\({\small\;\Rightarrow\left\{\textsf{通分分母を$10000$にすると、}\right\}\Rightarrow\;}\)\({\small\displaystyle\frac{1}{n}}{\displaystyle\sum_{i={\tiny\displaystyle\frac{5000}{10000}}n}^{{\tiny\displaystyle\frac{1250}{10000}}n}}\small cos^{2}\left({\scriptsize\displaystyle\frac{i\pi}{n}}\right)\)
\({\small\;=\;}{\small\displaystyle\frac{1}{n}}{\huge\lbrace}{\small cos^{2}\left({\scriptsize\displaystyle\frac{{\tiny\displaystyle\frac{5001n\pi}{10000}}}{n}}\right)}{\small\;+\;}{\small cos^{2}\left({\scriptsize\displaystyle\frac{{\tiny\displaystyle\frac{5002n\pi}{10000}}}{n}}\right)}{\small\;+\;}{\small cos^{2}\left({\scriptsize\displaystyle\frac{{\tiny\displaystyle\frac{5003n\pi}{10000}}}{n}}\right)}{\small\;+\;}\)\(\cdots\cdots{\small\;+\;}{\small cos^{2}\left({\scriptsize\displaystyle\frac{{\tiny\displaystyle\frac{1249n\pi}{10000}}}{n}}\right)}{\small\;+\;}{\small cos^{2}\left({\scriptsize\displaystyle\frac{{\tiny\displaystyle\frac{1250n\pi}{10000}}}{n}}\right)}\)\({\huge\rbrace}\)
初項 \({\small cos^{2}\left({\scriptsize\displaystyle\frac{{\tiny\displaystyle\frac{n\pi}{2}}}{n}}\right)}\)\({\small\;=\;}{\small cos^{2}\left({\scriptsize\displaystyle\frac{n\pi}{2n}}\right)}\)\({\small\;=\;}{\small cos^{2}\left({\scriptsize\displaystyle\frac{\pi}{2}}\right)}\)
末項 \(\small cos^{2}\left({\scriptsize\displaystyle\frac{\tiny\displaystyle\frac{7n\pi}{8}}{n}}\right)\)\({\small\;=\;}\small cos^{2}\left({\scriptsize\displaystyle\frac{7n\pi}{8n}}\right)\)\({\small\;=\;}\small cos^{2}\left({\scriptsize\displaystyle\frac{7\pi}{8}}\right)\)