Studying for Exams, here’s the problem.
Let ${a_n}, {b_n}$ be two sequences in $mathbb{R}$ and set
$A_n=sum_{k=0}^{n}a_k$.
Show the following
(a) $sum_{n=0}^{N}a_nb_n=sum_{n=0}^{N-1}A_n(b_n-b_{n+1})+A_Nb_N$
(b) If ${b_n}$ is decreasing and non-negative with $lim_{ntoinfty}b_n=0$, and if the sequence ${A_n}$ is bounded, then $sum_{n=0}^{infty}a_nb_n$ converges.
(c) If ${b_n}$ is decreasing and bounded below, and if $sum_{n=0}^{infty}a_n$ converges, then $sum_{n=0}^{infty}a_nb_n$ converges.
Here’s my thoughts so far.
(a) follows from induction.
(c) seems to follow from (b) if we let $c_n=b_n-lim_{ntoinfty}b$ (which exists since $b_n$) is monotone and bounded. If $sum_{n=0}^{infty}a_n$ converges, then the sequence of partial sums ${A_n}$ must be bounded, which gives the same conditions as (b).
I can’t get the solution for (b). We don’t have any information on non-negativity for ${a_n}$, so that shuts out a lot of routes for convergence. The form from (a) lends itself to check for some kind of telescoping but I haven’t made progress there. Is the Cauchy condition the way to go?
Any hints would be appreciated.
