Hmm it's just an equivalent of the ratio test :

if $\displaystyle \lim_{n \to \infty} ~ |a_n|^{1/n} < 1$ then it converges (I admit it is not the first formula the wikipedia gives)

for a), I have another way to propose to you.

You have $\displaystyle a_n=\left(\frac{1+\frac 1n}{3}\right)^n$

Noting that $\displaystyle n \geqslant 1$, you can say that $\displaystyle 1+\frac 1n \leqslant 2$ for all n.

So $\displaystyle a_n \leqslant \left(\frac 23\right)^n$

Does this series converge ?

Ooops, I'm sorry ! Actually, I guess it's better to use an alternating series test (I missed the (-1)^n part

)

In fact, I put this link so that you can see how they deal with such functions, how they conclude why it converges or diverges. It's similar here.

Excuse me in advance, but I don't know how to use the integral test, so maybe you can use it for one or two series here. Feel free to try