For any sequence s_n consider the arithmetic mean
t_n = (s1 + s2 + ... + sn) / n
Prove sn -> s implies tn -> s. Prove there are divergent sequences s_n which in this manner give rise to convergent sequences t_n.
If $\displaystyle s_{n} \rightarrow s$ we can set each $\displaystyle s_{k}= s + \delta_{k}$ where $\displaystyle \delta_{k} \rightarrow 0$. Now is...
$\displaystyle t_{n} =\sum_{k=1}^{n} \frac{s+\delta_{k}}{n} = s + \sum_{k=1}^{n} \frac{\delta_{k}}{n}$ (1)
... and because each term of the sum at the second term of (1) tends to 0 the sum itself tends to 0...
An example of divergent series that produces a convergent sequence $\displaystyle t_{n}$ is the armonic series. Ibn effect is...
$\displaystyle \sum_{k=1}^{n} \frac{1}{k} \sim \gamma + \ln n$ (2)
... where $\displaystyle \gamma$ is the 'Euler's constant'. From (2) You immediately derive that $\displaystyle t_{n} \rightarrow 0$...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
Given $\displaystyle \varepsilon>0$, there exists N such that $\displaystyle |s_k-s|<\varepsilon$ whenever k > N. Write $\displaystyle t_n$ as
$\displaystyle t_n-s = \frac{(s_1-s)+(s_2-s)+\ldots+(s_N-s)}n + \frac{(s_{N+1}-s)+(s_{N+2}-s)+\ldots+(s_n-s)}n $,
then show that the first of those fractions tends to 0, and the second fraction stays less than $\displaystyle \varepsilon$, as $\displaystyle n\to\infty.$
For the second part of the question, what about the sequence $\displaystyle s_n=(-1)^n$?