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Consider a telescoping series in the most general form

$$\sum_{n \geq n_0} (a_{n+k}-a_n)\,\,\,\,\,\,\,\, k\in \mathbb{N}$$

I found that this series:

  • Converges if $\mathrm{lim}_{n \to \infty} a_n= l \in \mathbb{R}$
  • If it converges, then the sum if equal to $a_1+a_2+...+a_k$

I would like to know if the previous statements are correct and also how can I find the partial sum of this series, i.e. what is the general expression of

$$\sum_{n=n_o}^{N} (a_{n+k}-a_n)=\,\,\,\,?$$

(If the previous is correct, then I should also have the following)

$$\mathrm{lim}_{N \to \infty} \sum_{n=n_o}^{N} (a_{n+k}-a_n)=a_1+a_2+...+a_k$$

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Let $k = 2$, and $a_0 = a_1 = 0$. Then let $$ a_2 - a_0 = \frac{1}{2} \\ a_3 - a_1 = \frac{1}{3} \\ a_4 - a_2 = \frac{1}{4} \\ \ldots $$ Your sum is then a part of the harmonics series, and hence diverges, even though $\lim a_n = 0 \in \mathbb R$.

So your first conjecture appears to be wrong, unless I've misread something.

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