Let $\displaystyle \sum_{n=1}^{\infty}a_n $ be a series of real numbers that converges then prove that:
- the series $\displaystyle \sum_{n=1}^{\infty} \frac{a_n}{n^x} $ converges uniformly on $[0, 1]$.
- the series $\displaystyle \sum_{n=1}^{\infty} \frac{a_n}{n+1}$ converges.
Questions: Does the series $\displaystyle \sum_{n=1}^{\infty} a_n $ need to converge absolutely in order for the conclusions to hold? May I have some hints for the first question? For the second one I tried this:
First of all I proved that the sequence goes to $0$ and then I tried the ratio test: $$\lim \left | \frac{\frac{a_{n+1}}{n+2}}{\frac{a_n}{n+1}} \right |=\left | \frac{a_{n+1}(n+1)}{a_n \left ( n+2 \right )} \right |<1 $$
because since the series $\displaystyle \sum_{n=1}^{\infty} a_n $ converges that means that $\displaystyle \lim \left | \frac{a_{n+1}}{a_n} \right |<1 $ and the other term goes to $1$. Is this OK?
For Abel's uniform convergence test, the series converge uniformly in $\left[0,1\right]$. About your question: no, the series doesn't need to converge absolutely, we have proved the uniform convergence without this hypothesis. And no, your proof isn't right because it could be $$\left|\frac{a_{n+1}}{a_{n}}\right|\rightarrow1$$ note that $$\left|\frac{a_{n+1}}{a_{n}}\right|\rightarrow L<1$$ works if the series converge absolutely. You can prove 2. using Dirichlet's test.