Convergence/Divergence of Series.

I am really poor in this part, been trying to do as much exercises as possible to get familiar with the patterns and I realised there are these few patterns that I couldn't seem to approach it.

Sometimes I know the right approach for it, but I have troubles simplifying the terms in order to prove that it fits certain criteria required to do conclusions. Here are a few cases, I hope that you guys would be able to give me a guidance on how to approach these different cases.

**Case I** : When there is root of something involved.

(i) $\displaystyle \sum^\infty_{n=1} \frac{1}{\sqrt{2n+1}}$

(ii) $\displaystyle \sum^\infty_{n=1} \frac{1}{\sqrt[3]{n^2 + 2}}$

(iii) $\displaystyle \sum^\infty_{n=1} \frac{1}{\sqrt[3]{3n^4 - 2}}$

**Case II** : When there is to the power of n.

(i)$\displaystyle \sum^\infty_{n=0} \frac {1 + 3^n}{1 + 4^n}$

(ii)$\displaystyle \sum^\infty_{n=1} \frac {3^n + 7n}{2^n(n^2 +1)}$

I reckon you approach this one with the ratio test? I always have troubles simplifying them!

**Case III** : When there is polynomials.

(i) $\displaystyle \sum^\infty_{n=1} \frac{n^3 + 4n}{n^4 + 200}$

(ii) $\displaystyle \sum^\infty_{n=1} \frac{5n^3 + 14n}{2n^6 + 200n^2 - 7}$

And the following few questions, I don't know how to classify them.

(i) $\displaystyle \sum^\infty_{n=0} (-1)^n$ - Alternating Series??

(ii) $\displaystyle \sum^\infty_{n=1} log (\frac {(n+1)^2}{n(n+2)})$

(iii) $\displaystyle \sum^\infty_{n=1} \frac{(n!)^2}{(2n)!}$ - I suppose you use Ratio test? But I have troubles simplifying this one!

Quite a number of questions there, I know! So sorry for the overload of questions :( Thanks for looking and helping though!!

Your help to master this topic would be greatly greatly appreciated!

Thank you so much!