\(\displaystyle{S=\lim_{n\rightarrow\infty}
\left(\dfrac{1}{n+1}+\dfrac{1}{n+2}+\dfrac{1}{n+3}+\cdots+\dfrac{1}{2n}\right)}\)
を求めよう。
\(S_n
=\dfrac{1}{n+1}+\dfrac{1}{n+2}+\dfrac{1}{n+3}+\cdots+\dfrac{1}{2n}\) とおく
数列 {xk} \(x_k=\dfrac{k}{n}\) (k=1,2,3,…, n)
数列 {wk} \(w_k=\dfrac{1}{n}\) (k=1,2,3,…, n)
\(f(x)=\dfrac{1}{1+x}\) とおくと
0≦ x1 < x2 < x3 <
… < xk < … < xn ≦ 1,
\(\displaystyle{\sum_{k=1}^nw_k=1}\)
\(\displaystyle{S_n=\sum_{k=1}^nf(x_k)w_k}\)
ゆえに，
\(\displaystyle{S=\lim_{n\rightarrow\infty}S_n}\)
\(\displaystyle{=\int_0^1\dfrac{1}{1+x}dx
=\left[\log(1+x)\right]_0^1=\log 2}\)