Convergence of a Sequence of Partial Sums

[AKJ1]

Homework Statement

Hi, I am reviewing a practice exam for my course and I am a bit stuck.

"Assume that a sequence of partial sums (s_n) converges, can we also then say the sequence a_n is convergent? Does this statement go both ways?

Answer: Yes, yes"

The Attempt at a Solution

On our exam, we merely pick yes or no for these type of questions, but I actually want to know why this is true. I can't find a counter example to say its false, but I don't actually know why its true.

My intuition:

A sequence of partial sums would be something like,

A1 , A1 + A2 , A1 + A2 + A3... So if this is convergent, the limit exists.

Now I am having difficulty directly following from this that the sequence a_n is also convergent.

S1 = A1
S2 = A1+A2
.
.
.
.
if we just apply the limit to both sides and show they both exist...is that enough? Is it possible one side will go to infinity?

In essence, for the limit of the partial sums to exist, we require that the terms of a_n decrease?

Thank you!

 

[BvU]

What is the meaning of "a sequence of partial sums (s_n) converges" ?
If that means $s_n \rightarrow l$ it also means $s_{n-1} \rightarrow l$, right ?
What is the consequence for $a_n$ ?

Theorem 3.2 http://web.mat.bham.ac.uk/R.W.Kaye/seqser/intro2series.pdf

 

[Fredrik]

Are you familiar with the concept of "Cauchy sequence"? It's pretty easy to prove the following implications:

$(s_n)$ is convergent $\Rightarrow$ $(s_n)$ is Cauchy $\Rightarrow$ $a_n\to 0$.

For the question of whether convergence of $(a_n)$ is sufficient to ensure convergence of $(s_n)$, consider the series $\sum_{n=1}^\infty\frac 1 n$.