Find the value of the unknown parameters so that the series converges How can I approach this exercise? 
Find all possible $a, b, c \in\mathbb{R}$ such that the series $\displaystyle \sum_{k=2}^{\infty} \frac{a^k}{k^b (lnk)^c}$ is convergent.
 A: Let 
$$
x_k = \frac{a^k}{k^b(\ln k)^c}
$$
denote the $k$th term of the series. This is going to be a bit of a whirlwind, so ask if anything is unclear!
The factor $a^k$ indicates that the ratio test may be a good place to start, and indeed, since
$$
\bigg|\frac{x_{k+1}}{x_k}\bigg| = |a|\cdot\bigg(\frac{k}{k+1}\bigg)^b\bigg(\frac{\ln k}{\ln(k+1)}\bigg)^c,
$$
we see that the series converges if $|a|<1$ and diverges if $|a|>1$. If $|a|=1$, the test is inconclusive.
There are now two cases: $a=1$ and $a=-1$.
If $a=1$, use the Cauchy Condensation Test. We get that the series converges if and only if the series
$$
\sum_{n=1}^\infty \frac{2^n}{2^{nb}n^c} = \sum_{n=1}^\infty \frac{2^{n(1-b)}}{n^c}
$$
converges. Using the ratio test, we see that the series diverges if $b < 1$ and converges if $b>1$. In the case where $b=1$, the series converges if $c>1$ and diverges if $c\leq 1$.
If $a=-1$, the series is an alternating series and will converge if and only if
$$
\lim_{k\to\infty} \frac{1}{k^b(\ln k)^c} = 0.
$$
I leave it to you to determine the criteria making this hold.
A: If you need to find all such $a,b,c$, then you can first show the series converges for $|a| < 1$ and diverges for $|a| > 1$, regardless of $b,c$. Then for $|a| = 1$, you can show the series converges if $b > 1$ and diverges for $b < 1$ if $a = 1$ regardless of $c$, and if $a = -1$, then the series converges if $b > 0$ regardless of $c$, and also converges if $b = 0$ and $c > 0$, otherwise diverges. Then you can show if $a = 1$ and $b = 1$ then the series converges if and only if $c > 1$.
I'm not sure what tools you have available but I think for the more subtle cases ($a = 1$) that Cauchy's condensation test would be very useful, but I don't know if you have that at your disposal so the above is a proof sketch for how to break into cases to prove (and also an outline of the final answer).
