Show that $𝑓_𝑛$ is in Hilbert space $H$ and that the series converges in H

To show that $f_n \in H$ you just need to prove that $\int_{-1}^1 (x^n)^2 dx < \infty$ for all $n$ (straightforward computation).

For the second question, you must note that you have a geometric progression. Each term is the previous one times $\alpha x$, where $-1<\alpha x < 1$. Hence the series converges to the formula you have written. Remember that in general the sum of a geometric series that grows by a factor of $|\beta|<1$ converges, and it converges to $\frac{c}{1-\beta}$, where $c$ is the first term of the sequence.

Finally, to show that also $f \in H$, you must check that $\int_{-1}^1 f(x)^2 dx < \infty$.