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prove that the series $(a_1-a_2)+(a_2-a_3)+(a_3-a_4)+...$ converges if and only if the sequence $\{a_n\}$ converges.
I know a series is convergent if the sequence of its partial sums is bounded. I do not know how to apply that to this question.

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  • $\begingroup$ What you wrote is not true. The sequence of partial sums of the sequence $(-1)^n$ is bounded but the series $\sum_{n=1}^\infty (-1)^n$ diverges. By definition a series converges if the sequence of partial sums converges. Well, what is the sequence of partial sums in your series? $\endgroup$ – Mark Oct 27 '19 at 23:48
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Let $s_n$ the partial sums of the series.

Then by telescoping $s_n=a_1-a_{n+1}$

So the series converges iff the sequence converges.

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  • $\begingroup$ @Bernard thank you for the correction. $\endgroup$ – Marios Gretsas Oct 27 '19 at 23:50
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Hints:

Note that the series you have is

$$\sum_{k=1}^\infty (a_k - a_{k+1})$$

It's partial sum, in turn, is

$$\sum_{k=1}^n (a_k - a_{k+1})$$

Consider trying a few small values for $n$, and consider the pattern that emerges. Use this to make a formula that is equivalent to the partial sum. In the context of that formula, what can you say happens to it as $n \to \infty$, both when $(a_n)$ is a divergent sequence, and when it is a convergent sequence?

(We approach this issue this way because we say $\sum_{k=1}^\infty b_k$ converges only if the sequence of partial sums - $\sum_{k=1}^n b_k$, for $n=1,2,3,\cdots$ - converges. Your definition of convergence for infinite sums isn't quite correct.)

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  • $\begingroup$ Thank you very much. Do you know of any textbook or resources so that I can study about this question in detail. I believe I know your note. but I can not see how to prove the sequence of partial sums is convergent if and only if the {a_n} sequence is convergent. $\endgroup$ – Shay Oct 28 '19 at 0:29

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