TITLE: Linear combination of series
QUESTION [0 upvotes]: Let $\{x_n\}_{n=1}^\infty$ be a sequence of real number. We know that if $\sum_{n=1}^\infty x_n$ converges, then $\sum_{n=1}^\infty \frac13 x_n-2x_{n+1}$ is convergent. 
My question is as follows. Suppose that  $\sum_{n=1}^\infty \frac13 x_n-2x_{n+1}$ converges. Can we conclude that $\sum_{n=1}^\infty x_n$ is convergent?

REPLY [0 votes]: Investigate the partial sums.
\begin{eqnarray}
T_n=\sum_{k=1}^n \frac{1}{3} x_k-2x_{k+1}&=&\frac{1}{3}\sum_{k=1}^nx_k-x_{k+1}-\frac{5}{3}\sum_{k=1}^nx_{k+1}
\\&=&\frac{1}{3}(x_1-x_{n+1})-\frac{5}{3}\sum_{k=2}^{n+1}x_{k}.
\end{eqnarray}
We know that $\lim_{n\to\infty} x_n= 0$ and $\lim_{n\to\infty} T_n= T$, since given series is convergent. Then taking the limit of both sides  we get $\sum x_k$ is convergent.
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