I'm having trouble with a practice problem in sequences and series and would like some guidance. The problem is:
Given a series of real numbers $\sum_{k=1}^\infty x_k$, let $\sum_{n=1}^\infty y_n$ be another series of real numbers such that :
$y_1 = x_1$
$y_2 = \frac{x_2}{2}$
$\vdots$
$\displaystyle y_n = \frac{1}{n(n-1)}x_2 + \frac{2}{n(n-1)}x_3 + \ldots \frac{n-2}{n(n-1)}x_{n-1} + \frac{1}{x_n}$
or, more concisely:
$\displaystyle y_n = \sum_{k=2}^n\frac{(k-1)}{n(n-1)}x_k$
The problem asks to prove if $\sum_{k=1}^\infty x_k$ converges, then $\sum_{n=1}^\infty y_n$ also converges $\textit{to the same value}$.
I can easily use the comparison test to prove that if one series converges, then the other must converge as well. However, I have no idea how to go about showing that both series converge to the same value.
One unsuccessful approach that I tried was to show that the partial sum $s_n = \sum_{k=1}^n$ and $t_n = \sum_{k=1}^n y_k$ were equivalent for all $n$. In the process of attempting this, I think that I actually managed to show that they were not....
This makes sense because my Prof. also gave us a hint saying that the second series may converge even though the first may not, and that the result can be used to say something along the lines of:
$\displaystyle 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 \ldots = \frac{1}{2}$
which makes absolutely no sense at all.