Show that $$\sum_{k=1}^{\infty}e^{kx} = e^x+e^{2x}+\cdots$$ is uniformly convergent for $|x|<\frac14.$
I tried something like:
We have the $n$th partial sum as a function, $$S_n(x)= \sum_{r=1}^n e^{rx}$$
and so $$S_n(x)=\frac{e^x(1-e^{nx})}{1-e^x}.$$
Now, $$S(x)= \lim_{n\to\infty} S_n(x)=\frac {e^x}{1-e^x}$$ when $x\ne 0$ and when $x=0$ then $S(x)=1$.
Let for given $\epsilon>0$ there exist $N$ such that $|S_n(x)-S(x)|<\epsilon$ for all $n\geq N$. We obtain $\left|\frac{e^x}{1-e^x} (-e^{nx})\right|$ here.. How can I proceed or some other way to solve?
This series doesn't pass the divergence test. The individual terms don't go to zero. $e^{rx}\stackrel{r\to\infty}\to \infty $, for $0\lt x\lt\dfrac 14$. So the series diverges.