# $a_n=(-1)^{n-1}, \; s_n=\sum_{i=1}^{n}a_i$ then find $\lim_{n\to \infty}\frac{s_1+s_2+\dots s_n}{n}$

$$a_n=(-1)^{n-1}, \; s_n=\sum_{i=1}^{n}a_i$$ then find $$\lim_{n\to > \infty}\frac{s_1+s_2+\dots s_n}{n}$$

$$s_k=1,\; \text{if k is odd and } s_k=0 \text{ if k is even}$$

Cauchy's theorem for a sequence $$(x_n)$$ in $$R$$, we have $$\lim\frac{x_1+x_2+\dots x_n}{n}=\lim x_n$$

How do I make use of this theorem here when $$(s_n)$$ is oscillating between $$0$$ and $$1$$?

• Consider the $\limsup$ and $\liminf$ of $\frac{s_1+\cdots+s_n}n$. – Math1000 Feb 5 at 20:10

## 1 Answer

I believe that the theorem you are referring to says:

If a sequence of real numbers $$\{x_n\}_{n=1}^\infty$$ converges to $$x \in \mathbb R$$, then $$\lim_{n\to \infty} \frac{x_1+x_2+\cdots+x_n}n=x$$.

The sequence $$\{s_n\}_{n=1}^\infty$$ however does not converge as your computations suggest. But if we define $$c_n:=\frac{s_1+s_2+\cdots+s_n}n$$, then we have $$c_n=\frac12$$ whenever $$n$$ is even and $$c_n=\frac12+\frac1{2n}$$ whenever $$n$$ is odd. Therefore $$\lim_{n \to \infty} c_n=\lim_{n \to \infty}\frac{s_1+s_2+\cdots+s_n}n=\frac12$$ as both of these aforementioned subsequences clearly converge to $$\frac12$$.

The sequence is an example of why the converse of the theorem in question is not true.