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Suppose the sequence $\{c_n\}$ converges to zero. Prove $\sum_{n \geq 0}c_n$ converges iff $\sum_{k \geq 0}(c_{2k}+c_{2k+1})$ converges. Moreover, if the two series converge then they have the same limit.

I was thinking that if $$\lim_{n \rightarrow \infty}c_n \neq 0$$ Then the statement can't be true. Like the sequence $$1, -1, 1 ,-1 , 1 ,-1, \cdots$$ So $$\sum_{k \geq 0}(c_{2k}+c_{2k+1})=0$$ which is convergent, but $$\sum_{n \geq 0}c_n \text{does not exists}$$ So how could we prove under the assumption $\lim_{n \rightarrow \infty}c_n=0$? Many thanks~

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If $\sum_{n=0}^{\infty}c_n = a$ then we have $$S_n = \sum_{k=0}^n c_k \to a.$$ So, $$ \sum_{k=0}^n (c_{2k} + c_{2k+1}) = S_{2n+1} \to a. $$

I'll let you handle the (slightly messier) other direction.

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    $\begingroup$ And why don't we say $\sum_{k=0}^{n}(c_{2k}+c_{2k+1})=s_{2n+1}$ directly? $\endgroup$ – Nan Jan 15 '18 at 1:51
  • $\begingroup$ @Nan good point, I just messed up. (The other direction will make essential use of $c_n\to 0.$) $\endgroup$ – spaceisdarkgreen Jan 15 '18 at 2:33

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