First slide

Introduction to definite integration

Question

If $S_{n} = \frac{e^{1 / n}}{n^{2}} + \frac{2}{n^{2}} e^{2 / n} + \frac{3}{n^{2}} e^{3 / n} + \dots + \frac{1}{n} e$ then $lim_{n \to \infty} S_{n}$ is

Easy

A

1
B

$e$
C

0
D

none of these

Solution

$\begin{array}{l} S_{n} = \frac{1}{n} \sum_{k = 1}^{n} \frac{k}{n} e^{k / n} \\ lim_{n \to \infty} S_{n} = \int_{0}^{1} x e^{x} d x = 1 . \end{array}$

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