If $\sum a_n$ is a convergent series and $S=\lim{s_n}$ where $s_n$ is the $n$th partial sum, how would I prove that $\lim_{n\to \infty} \frac{s_1+...+s_n}{n} = S$?
I'm not sure where to start, I've been having trouble understanding partial sums. Any tips on where to begin?
See my answer (or other answers) at If $\sum a_n$ is a convergent series with $S = \lim s_n$, where $s_n$ is the nth partial sum, then $\lim_{n \to \infty} \frac{s_1+...+s_n}{n} = S$
In reference to your question about understanding partial sums, the partial sum for the series $\sum a_k$ is defined as $$ s_n = \sum_{k=1}^na_k.$$
So $$s_1 = a_1\\s_2=a_1+a_2\\s_3 = a_1+a_2+a_3$$
etc. So the partial sum for the series is what you'd get if you added it up and stopped at $n.$
We take $n\rightarrow \infty$ in the formula above and write $$ \lim_{n\rightarrow\infty}s_n = \sum_{k=1}^\infty a_k = a_1+a_2+a_3+\ldots$$ so the sum of the infinite series is defined by the limit of the sequence of partial sums, if the limit exists.
The limit in your problem is $$\lim_{n\rightarrow\infty} \frac{s_1+s_2+\ldots s_n}{n}$$ which you can think of as the "long run average" of the sequence of partial sums. So you are asked to show that if the sequence $s_n\rightarrow S$ then the long-run average is equal to $S.$ (This is actually true for any convergent sequence... the fact that this sequence comes from the partial sum of a series is immaterial.)
A key fact is that sometimes the "long-run average" of a sequence exists even when the sequence does not converge (can you think of an example?). This is the origin of the definition of Cesaro summability. What you're proving here is that if a series converges, its Cesaro sum exists and has the same value.
This problem has two parts: 1. The definition of a convergent infinite series; 2. Basic theorem of Cesaro: If $x_n \to L,$ then $(x_1 + \cdots + x_n)/n \to L.$
Suppose that $\sum_{n=1}^{\infty}a_n= S.$ By definition, this means $S_n \to S,$ where $S_n=\sum_{k=1}^{n}a_k.$ By Cesaro, this implies $(S_1 + \cdots + S_n)/n \to S.$
So I haven't proved Cesaro, which has been proved on MSE approximately $734$ times. But I thought it was worthwhile pointing out that the series result here, after you see through the incense and incantations, is just Cesaro's basic result on sequences.
