次の4つの極限を求めます。 (1) $\lim_{n \to \infty} (\frac{1^4}{n^5} + \frac{2^4}{n^5} + \dots + \frac{n^4}{n^5})$ (2) $\lim_{n \to \infty} (\frac{n+1}{n^2} + \frac{n+2}{n^2} + \dots + \frac{n+n}{n^2})$ (3) $\lim_{n \to \infty} (\frac{e^{\frac{1}{n}}}{n} + \frac{e^{\frac{2}{n}}}{n} + \dots + \frac{e^{\frac{n}{n}}}{n})$ (4) $\lim_{n \to \infty} (\frac{\cos \frac{\pi}{2n}}{n} + \frac{\cos \frac{2\pi}{2n}}{n} + \dots + \frac{\cos \frac{n\pi}{2n}}{n})$

1. 問題の内容

次の4つの極限を求めます。

(1) $\lim_{n \to \infty} (\frac{1^4}{n^5} + \frac{2^4}{n^5} + \dots + \frac{n^4}{n^5})$

(2) $\lim_{n \to \infty} (\frac{n+1}{n^2} + \frac{n+2}{n^2} + \dots + \frac{n+n}{n^2})$

(3) $\lim_{n \to \infty} (\frac{e^{\frac{1}{n}}}{n} + \frac{e^{\frac{2}{n}}}{n} + \dots + \frac{e^{\frac{n}{n}}}{n})$

(4) $\lim_{n \to \infty} (\frac{\cos \frac{\pi}{2n}}{n} + \frac{\cos \frac{2\pi}{2n}}{n} + \dots + \frac{\cos \frac{n\pi}{2n}}{n})$

2. 解き方の手順

(1) $\lim_{n \to \infty} (\frac{1^4}{n^5} + \frac{2^4}{n^5} + \dots + \frac{n^4}{n^5}) = \lim_{n \to \infty} \sum_{k=1}^n \frac{k^4}{n^5} = \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n (\frac{k}{n})^4$

これは定積分 $\int_0^1 x^4 dx$ で表せる。

$\int_0^1 x^4 dx = [\frac{x^5}{5}]_0^1 = \frac{1}{5}$

(2) $\lim_{n \to \infty} (\frac{n+1}{n^2} + \frac{n+2}{n^2} + \dots + \frac{n+n}{n^2}) = \lim_{n \to \infty} \sum_{k=1}^n \frac{n+k}{n^2} = \lim_{n \to \infty} \sum_{k=1}^n (\frac{1}{n} + \frac{k}{n^2}) = \lim_{n \to \infty} (\sum_{k=1}^n \frac{1}{n} + \sum_{k=1}^n \frac{k}{n^2})$

$= \lim_{n \to \infty} (1 + \frac{n(n+1)}{2n^2}) = \lim_{n \to \infty} (1 + \frac{1}{2} + \frac{1}{2n}) = 1 + \frac{1}{2} = \frac{3}{2}$

(3) $\lim_{n \to \infty} (\frac{e^{\frac{1}{n}}}{n} + \frac{e^{\frac{2}{n}}}{n} + \dots + \frac{e^{\frac{n}{n}}}{n}) = \lim_{n \to \infty} \sum_{k=1}^n \frac{e^{\frac{k}{n}}}{n} = \int_0^1 e^x dx$

$\int_0^1 e^x dx = [e^x]_0^1 = e^1 - e^0 = e - 1$

(4) $\lim_{n \to \infty} (\frac{\cos \frac{\pi}{2n}}{n} + \frac{\cos \frac{2\pi}{2n}}{n} + \dots + \frac{\cos \frac{n\pi}{2n}}{n}) = \lim_{n \to \infty} \sum_{k=1}^n \frac{\cos \frac{k\pi}{2n}}{n} = \int_0^1 \cos(\frac{\pi}{2} x) dx$

$\int_0^1 \cos(\frac{\pi}{2} x) dx = [\frac{2}{\pi} \sin(\frac{\pi}{2} x)]_0^1 = \frac{2}{\pi} (\sin \frac{\pi}{2} - \sin 0) = \frac{2}{\pi} (1 - 0) = \frac{2}{\pi}$

3. 最終的な答え

(1) $\frac{1}{5}$

(2) $\frac{3}{2}$

(3) $e-1$

(4) $\frac{2}{\pi}$