Converge the sequence $\left(\left(1+\frac{1}{n}\right) \left(1+\frac{2}{n}\right)\cdots\left(1+\frac{n}{n}\right)\right)^{1/n}$

Question

Does this series converge or diverge?

$$ \left[\prod_{k=1}^{n}\left(1+\frac{k}{n}\right)\right]^{\frac{1}{n}} $$

Here's another way of evaluating it without using Reimann Sums. — r9m, Apr 09 '15 at 00:53
To evaluate the limit, see the link above; it has been asked before. Now, if you only need to check whether it converges or not, but not where, you can start by observeing that it is definitely bounded by $1$ and $2$. Can you say anything about its monotonicity? — Theo Douvropoulos, Apr 09 '15 at 01:10

score 1 · Answer 1 · answered Apr 09 '15 at 00:50

1

Hint This sequence is, up to an exponential, equivalent to:

$$\sum_{k=1}^{n}\frac{1}{n}\ln\left(1+\frac{k}{n}\right) =\int_{1}^2\cdots\cdots$$

(use Riemann sums)

answered Apr 09 '15 at 00:50

Elaqqad

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It looks like you applied the natural logarithm (and its rules) to the above expression to end up with a Riemann sum. Is this correct? – wjmolina Apr 09 '15 at 01:06
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Yes this correct – Elaqqad Apr 09 '15 at 01:08

1 Answers1