let $$s(x)=\sum_{k=0}^{\infty}\ x^k$$
I have to show that the series converges to $$\frac{1}{1-x}$$ for $$|x|<1$$
I can easily show the series converges for $|x|<1$ using the Ratio test. But how do I show it converges to $1/1-x$?
After that, I have to show that $s^2(x)=\sum_{k=0}^{\infty}\ (ak) x^k$ using the cauchy product. so what I wanted to do is multiply $\sum_{k=0}^{\infty}\ x^k$ by $\sum_{k=0}^{\infty}\ x^k$ which should give me the result $\sum_{k=0}^{\infty}\ ck$ which is equal to $\sum_{k=0}^{\infty}\ \sum_{m=0}^{k}\ x^m.x^k-^m)$
What should I do next? or am I completely wrong?
Take sequence of partial sums:
$$s_n=\sum_{k=0}^nx^k=\frac{1-x^n}{1-x}\xrightarrow[n\to\infty]{}\frac1{1-x}\;,\;\;\text{since}\;\; |x|<1\implies x^n\to 0$$