問題
極限値$\displaystyle\lim_{n \to \infty}\displaystyle\frac{1}{n^2}(e^\frac{1}{n}+2e^\frac{2}{n}+3e^\frac{3}{n}+\cdots+ne^\frac{n}{n})$を求めよ.
【岩手大学 2012】
解答
$\displaystyle\lim_{n \to \infty}\displaystyle\frac{1}{n^2}(e^\frac{1}{n}+2e^\frac{2}{n}+3e^\frac{3}{n}+\cdots+ne^\frac{n}{n})$
$=\displaystyle\lim_{n \to \infty}\displaystyle\frac{1}{n}(\displaystyle\frac{1}{n}e^\frac{1}{n}+\displaystyle\frac{2}{n}e^\frac{2}{n}+\displaystyle\frac{3}{n}e^\frac{3}{n}+\cdots+\displaystyle\frac{n}{n}e^\frac{n}{n})$
$=\displaystyle\lim_{n \to \infty}\displaystyle\frac{1}{n}\displaystyle\sum_{k=1}^{n}\displaystyle\frac{k}{n}e^{\frac{k}{n}}$
$=\displaystyle\int_{0}^{1}xe^xdx$
$=[xe^x]_{0}^{1}-\displaystyle\int_{0}^{1}e^xdx$
$=e-[e^x]_{0}^{1}$
$=e-(e-1)$
$=1$
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