What series test would you use on these and why? I just started learning series , I am trying to put everything together...I have few some few random problems just to see what kind of strategy you would use here...


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

*$\displaystyle\sum_{n=1}^\infty\frac2{(2n - 1)(2n + 1)}$;  telescoping series ?

*$\displaystyle\sum_{n=1}^\infty\frac1{n(1 + \ln^2 n)}$

*$\displaystyle\sum_{n=1}^\infty\frac1{\sqrt{n (n + 1)}}$;   integral test ? cause you could do u-sub?

#1 is just teeming with $n$th powers. Use the root test. #2 and #4 can be compared with $\sum 1/n^p$ for some $p$ (limit comparison). For #3 I would compare with $$\sum\frac{1}{n\ln n}$$ (which diverges by a standard application of the integral test).
