How can I show that the series $\sum _{k =1}^{\infty} \frac{1}{k (1 + (1/k))^k}$ is divergent.

Question

How can I show that the series $\sum\limits_{k =1}^{\infty} \frac{1}{k \left(1 + \left(\frac{1}{k}\right)\right)^k}$ is divergent?

I know that I will compare it with the divergent harmonic series, but how is the harmonic series smaller than this?

Answer 1

Since $(1+\frac{1}{k})^k \leq e < 3$ $\forall k$ we can infer that your series is bounded below by $\sum \frac{1}{3n}$ which diverges to $\infty$ (the harmonic series).

More generally, one can use the following sometimes called comparison test for series: if $a_n$, $b_n$ are positive sequences such that $\lim \frac{a_n}{b_n}$ is finite and non-zero, then $\sum a_n$ converges iff $\sum b_n$ does.

Answer 2

Hint:

We have $\;\Bigl(1+\frac 1 k\Bigr)^k<\mathrm e$.