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How to prove convergence of the following?

$$\sum_{n=1}^\infty \left(\frac{n}{n+1}\right)^n $$

Is the following statement true? $$\sum_{n=1}^\infty \left(\frac{n}{n+1}\right)^n < \infty$$

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Here is an answer to a similar problem that gets a bit more to the core of the answer: math.stackexchange.com/a/597781/3584 –  eBusiness Dec 8 '13 at 9:31
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4 Answers 4

up vote 5 down vote accepted

It does not converge. Look at $$ \lim_{n\to \infty} \left(\frac{n}{n+1}\right)^n=\frac{1}{e}\neq 0 $$ So the series cannot converge.

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So, this series is divergent. $$\sum_{n=1}^∞ {\left({\frac{n}{n+1}}\right)}^n = \infty?$$ –  user110037 Dec 7 '13 at 21:57
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@user110037 Yes, this is by what is called here the 'limit of the summand' test. It is usually the first one you want to do because it is a simple task to do and easily checks to see if something is necessarily divergent. Though if the limit is $0$, it could still be divergent, take the harmonic series! Anyway, your series again is divergent by the first test listed here: en.wikipedia.org/wiki/Convergence_tests –  mathematics2x2life Dec 7 '13 at 21:59
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Hint: What is the limit of the sequence $\left(\left(\dfrac{n}{n+1}\right)^n\right)_{n\in \Bbb N}$?

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Note that $$\left(\dfrac{n}{n+1}\right)^n = \dfrac1{\left(1+\dfrac1n\right)^n} \sim \dfrac1{e}$$ and conclude what you want.

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@GitGud Ok. Stupid error. Thanks. Corrected. –  user17762 Dec 7 '13 at 21:52
    
All the cowards can now remove their down-votes. –  user17762 Dec 7 '13 at 21:54
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-1 For calling me a coward. And for worrying more about speed than correctness of answer. –  eBusiness Dec 7 '13 at 22:03
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@eBusiness Ok. Thanks for identifying yourself. –  user17762 Dec 7 '13 at 22:05
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If a series $\sum_{n=1}^\infty a_n$ converges, then $a_n\to 0$. Which is equivalent to saying that if $a_n$ does not tend to zero, then the corresponding series does not converge.

So, it suffices here to observe that $(\frac{n}{n+1})^n\to\frac{1}{e}$.

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