First slide

Introduction to definite integration

Question

$The value of lim_{n \to \infty} \sum_{K = 1}^{n} \frac{K}{n^{2} + K^{2}}$

Moderate

A

$\log 2$
B

$\log \sqrt{2}$
C

log4
D

$\log \sqrt{3}$

Solution

$\begin{array}{l} \begin{matrix} \lim_{n \to \infty} \sum_{k = 1}^{n} \frac{K}{n^{2} + K^{2}} & = & \lim_{n \to \infty} \sum_{k = 1}^{n} \frac{K}{n^{2} + K^{2}} \end{matrix} \\ = \lim_{n \to \infty} \sum_{k = 1}^{n} \frac{1}{n^{2}} \times \frac{K}{1 + {(\frac{K}{n})}^{2}} \\ = \lim_{n \to \infty} \sum_{k = 1}^{n} \frac{1}{n} \times \frac{K / n}{1 + {(\frac{K}{n})}^{2}} \\ = \int_{0}^{1} \frac{x}{1 + x^{2}} d x \\ = {[\frac{1}{2} \log (1 + x^{2})]}_{0}^{1} \\ = \frac{1}{2} \log 2 \end{array}$

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