which of the following series converges uniformly for $x\in (-π,π)$.
1.$\sum_{n=1}^\infty \frac{x^n}{n^n}$.
2.$\sum_{n=1}^\infty \frac{1}{((x+π)n)^2}$
I was trying to put putting $x = 0$,then both are uniformly convergent.I'm confused. Please give me hints or any idea, or tell me the solution.I would be much thankful.
For part (1) you have $$ \left|\frac{x}{n}\right|^n< \left|\frac{\pi}{n}\right|^n$$ and $$\sum_{n=1}^\infty \left|\frac{\pi}{n}\right|^n$$ converges so the M-test applies here.
Part (2) is funny, cause it seems pretty benign: $$ \sum_{n=1}^\infty \frac{1}{((x+\pi)n)^2} = \frac{1}{(x+\pi)^2}\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6(x+\pi)}$$ and yet it doesn't uniformly converge since whatever small difference remains from the nicely converging $\sum \frac{1}{n^2}$ is blown up arbitrarily large by the divergence as $x\to -\pi$