Key to "Show whether the series converges.∑n=1∞∫01nex−11+xdx"

Carole Juarez

Answered question

2022-02-23

Show whether the series converges.

\sum_{n = 1}^{\infty} \int_{0}^{\frac{1}{\sqrt{n}}} \frac{e^{x} - 1}{1 + x} d x

Alexandra Haynes

Beginner2022-02-24Added 10 answers

For

n > 0

, we have

0 \leq x \leq 1

thus

e^{x} - 1 \geq x

and

\frac{1}{1 + x} \geq \frac{1}{2}

from this

\int_{0}^{\frac{1}{\sqrt{n}}} \frac{e^{x} - 1}{1 + x} d x \geq \frac{1}{2} {[\frac{x^{2}}{2}]}_{0}^{\frac{1}{\sqrt{n}}}

\geq \frac{1}{4 n}

the series is Divergent.

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