This is a simple series, it was listed in the review section of my book, they state for this problem determine if the series converges or diverges by using the integral test, however, I feel limiting myself to one method of showing convergence will not help me on the test, so I need someone to check my different ways of showing this converges

The initial problem $\displaystyle \sum_{n=1}^{\infty}\frac{1}{(n+2)^2}$

Using the integral test

$\displaystyle \sum_{n=1}^{\infty}\frac{1}{(n+2)^2} = \lim_{b\to\infty}\int_1^b \frac{1}{(x+2)^2} dx$

Using some simple substitution to find the integrand of

$\displaystyle

\lim_{b\to\infty}- \frac{1}{(x+2)} |_1^b$

Then

$\displaystyle

\lim_{b\to\infty}- \frac{1}{(b+2)} + \lim_{b\to\infty}\frac{1}{(3)}=\frac{1}{3}$

Since the limit is finite the series converges.

Using a direct comparison

$\displaystyle \sum_{n=1}^{\infty}\frac{1}{(n+2)^2}\le\sum_{n=1}^ {\infty}\frac{1}{n^2}$

We know by the p-series test for $\displaystyle p>1$ converges, thus the series converges

We can also take the limit of the series itself and apply L'Hopital's rule and see the limit is 0 but this doesn't mean much, because it could also diverge or converge