Evaluation of given limit where $ n \to \infty$

Asked May 08 '17 at 04:37

Active May 08 '17 at 05:08

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Evaluate the given limit:

$$ \lim_{n \to \infty} \bigg[ \bigg(\frac{n}{n} \bigg)^n +\bigg(\frac{n-1}{n} \bigg)^n + \bigg(\frac{n-2}{n} \bigg)^n+....+\bigg(\frac{1}{n} \bigg)^n \bigg]$$

I was trying to calculate each limit separately. First bracket equals $1$, second bracket equals $1/e$ , third bracket equals $1/e^2$ and so on. But how to add up after that? Could someone help me with this?

edited May 08 '17 at 05:02

asked May 08 '17 at 04:37

Mathematics

3,973

false. the third bracket is 1/e^2 – Saketh Malyala May 08 '17 at 04:52
@SakethMalyala Yes. Thanks for pointing it out. – Mathematics May 08 '17 at 05:03
1

One can take limits term by term limits via monotone convergence theorem to get the answer as $e/(e-1)$. – Paramanand Singh May 08 '17 at 05:39

Evaluation of given limit where $ n \to \infty$

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