Originally Posted by

**iLikeMaths** I have been having trouble with this question for days and i am having difficulty understanding what the question is asking me, its on the subject of series convergence and divergence, its a long question

This question concerns grouping the terms of a series as follows:

a0+(a1+a2)+(a3+a4)+(a5+· · ·+a8)+(a9+· · ·+a16)+(a17+· · ·+a32)+· · ·

Suppose that $\displaystyle 0<a_{n+1}<a_{n}$ for all $\displaystyle n$.

Show that

$\displaystyle \frac {1}{2}(2^{n+1} a_{2n+1})<a_{2n+1}+....<2^n a_{2n}$

hence show that if this series

$\displaystyle \sum a_n$ converges then this series $\displaystyle \sum 2^n a_2n$ also converges. and that if one of the two series diverges then so does the other.

hence determine if these series converge or diverge

$\displaystyle \sum \frac {1}{n \log n}$ and $\displaystyle \sum \frac {1} {n(\log n)^2}$