Thread: converging or diverging series

Do the following series converge or diverge?
Sum(1/(k^(1/3)) k=0..infinity

Sum(1/(k*log(k)) k=2..infinity

What is the sum (exact form) of
Sum(k*p^(k-1)) k=0..infinity and |p|<1

^ is power to
* is multiplication

[Difficulty]
For the first two I tried the Ratio and Root test and found L=1 in
both cases. Also the divergence test is inconclusive. How do I
determine if the series converges or diverges?

I recommend the Condensation Test. If a_n is a decreasing sequence then Sum a_n converges or diverges with Sum 2^n a_{2^n}.
The proof is to bracket terms (a_1) + (a_2) + (a_3+a_4) + (a_5+...+a_8) + (a_9+...+a_16) + ... and compare.

In this case, condensation shows that Sum n^{-s} converges or diverges with Sum 2^n 2^{-ns} which converges if s>1 and diverges if s<=1. Applying it to Sum 1/(n log n) we consider Sum 2^n / (2^n n log 2) which we just saw diverges.

Another useful test in this case is the Integral Test. Compare Sum f(n) with Int f(x) dx, assuming you can do the integral!

Finally, consider Sum k x^{k-1}. It looks like the derivative of something ...