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Does $S = \sum_{n=1}^{\infty}(-1)^n (e- (1+ \frac{1}{n})^n)$ converge or diverge?

My attempt : I know that $e = \lim_{n\rightarrow \infty}( 1+ \frac{1}{n})^n$.

Now put the value e in given series $S$ , I got $\sum_{n=1}^{\infty}(-1)^n (e- (1+ \frac{1}{n})^n)= \sum_{n=1}^{\infty}(-1)^n (e- e)=0$

so the given series is converges

is it correct????

Bernard
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jasmine
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  • Personally I would be looking to use the alternating series test for this. You should work out the first few terms to get a sense of what is happening - though the general term tends to zero, each term is itself constant and doesn't vary with $n$ – Mark Bennet Oct 31 '18 at 10:44

1 Answers1

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The series converges, but not for the reason you state.

It is known that $\left(1+\frac1n\right)^n$ converges to $e$ monotonically. Therefore $e-\left(1+\frac1n\right)^n$ is a sequence monotonically decreasing to $0$ and Leibniz criterion applies.

Bernard
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Hagen von Eitzen
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