# Thread: Testing for Convergence of Series

1. ## Testing for Convergence of Series

Hello, and thanks for help in advance. I have an exam tomorrow that I'm very nervous about, so here goes:

I understand the justification for quite a few of the tests for convergence or divergence of series, such as the Comparison & Integral tests, but often have trouble applying any of them.

One example is the following:

If from n = 1 to n = infinity, is 1/(n + 4)^(1/2) convergent or divergent?

I understand that for p-series 1 / n^p, the function converges where p > 1 and diverges otherwise. Can I then say, because

1/(n + 4)^(1/2) ~ 1/(n)^(1/2) where 1/(n)^(1/2) diverges

that my original series (on left) diverges?

Are approximation methods like this (whether or not this one specifically works) allowed?

2. Hello,

You can, but you can also do it this way :

change the indice :

$\sum_{n=1}^\infty \frac{1}{(n+4)^{1/2}}=\sum_{n=5}^\infty \frac{1}{n^{1/2}}<\sum_{n=1}^\infty \frac{1}{n^{1/2}}$

3. That is a good way of justifying the ~.

Are there any examples that make simplification via a change of the indice more understandable?