What is the Convergence of the Infinite Sum with k^(1/k)?

[Quinzio]

Homework Statement

Show that
$$\sum_{k=0}^{\infty} \sqrt[k]k-1$$

converges.

Homework Equations

Ratio, radix theorems, comparison with other sums...

The Attempt at a Solution

No idea whatsoever.
Where does one begin in this case ? With other cases I'm quite confident.

 

[Dickfore]

Hint:
Let the general term be denoted by:
$$a_k = \sqrt[k]{k} - 1$$
Do you know how to prove that $\lim_{k \rightarrow \infty}{a_k} = 0$?
Then, we have:
$$k = (1 + a_k)^{k}$$
Taking the same equality for $k + 1$, and subtracting this one, you ought to get:
$$1 = (1 + a_{k + 1})^{k + 1} - (1 + a_k)^{k}$$
Solve this equation for $a_{k + 1}$. What do you get?

Because as $k \rightarrow \infty$, $a_k$ is an infinitesimal quantity, you may expand your expression for $a_{k + 1}$ in powers of $a_{k}$ up to the first non-vanishing order. What do you get?

The solution for this asymptotic recursion relation would give you a comparison general term $b_n$, and the series $\sum_{n = 1}^{\infty}{b_n}$ is pretty easy to test for convergence. Then, you may use the [STRIKE]Ratio comparison test[/STRIKE].

EDIT:
Use the Asymptotic comparison test mentioned here instead.

 

[Quinzio]

I don't quite follow you.

I sort of need simpler methods, thanks anyway.

 

[Quinzio]

Yep, asymptotic comparison seems a good way.