# prove this series is convergent

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.

• 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? – Mark Oct 27 '19 at 23:48

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.

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

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.)

• 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. – Shay Oct 28 '19 at 0:29