if ak>0 and sum(ak) converges. will the series sum((ak)^(1/2)) converge

Hello fmnushi!

The answer is that it will not always converge. Take for example the
sequence

ak = 1/(k^2)

then sum(ak) (with k from 0 to infinity) converges. You can check this
result in the page that follows. However, ak^(1/2) gives 1/k. The
sum(1/k) (which is called the harmonic series) is not convergent. This
result is always shown in the following link.

Convergence Tests for Infinite series
http://www.math.hmc.edu/calculus/tutorials/convergence/


Google search strategy used
series convergence p-series
://www.google.com/search?q=series+convergence+p-series&hl=en&lr=&ie=UTF-8&oe=UTF-8&start=10&sa=N


I hope this helps! If you have any doubts regarding my answer, please
don't hesitate to request a clarification.


Best wishes!
elmarto