These should be quick tasks: We know that the Legendre Polynomials satisfy $\int_{-1}^{1} P_m(x)P_{n}(x)dx= \delta_{mn}\frac{2}{2n+1}$
Which of the following series converge ( $ \forall x \in [-1,1]$ )?
i) $\sum_{n=0}^\infty (-1)^nP_n(x)$
ii) $\sum_{n=0}^\infty n^2 2^{-n} P_n(x)$
iii) $\sum_{n=0}^\infty P_{n^2}(x)$
iv) $\sum_{n=0}^\infty \frac{(-1)^n}{\sqrt{n}}P_n(x) $
I tried with the Parseval Identity $||f||^2=\sum_{n=0}^\infty |(P_n,f)|^2$ but I didn't find a useful $f$ to plug in and somehow get any of the sums on the right hand side.
I think the 2nd sum does converge since we have exponential decrease and $P_n$ is a polynomial. Since iv) grows slower than i) I assume iv) converges and i) doesn't, but I don't have any real mathematical argument for that.