Give examples of a formal power series $\sum_{n=0}^\infty c_n x^n$ centered at $0$ with radius of convergence $1$, which
(a) diverges at both $x=1$ and $x=-1$;
(b) diverges at $x=1$ but converges at $x= -1$;
(c) converges at $x=1$ but diverges at $x=-1$;
(d) converges at both $x =1 $ and $x =-1$;
(e) converges pointwise on $(-1, 1)$, but does not converges uniformly on $(-1, 1)$.
For (a), $c_n$ can be any constant real number or $c_n = n$ also works.
For (b), $c_n = \frac1n$ works.
For (c), $c_n = (-1)^n \frac1n$ works.
I have difficulty in coming up with examples for (d) and (e). Can you give me some hint?
$\sum \frac 1 {n^{2}} x^{n}$ has radius of convergence $1$ and it converges at both $x=1$ and $x=-1$.
$\sum n x^{n}$ converges for $|x|<1$ but the convergence is not uniform since the general term does not tend to $0$ uniformly (as seen by putting $x=\frac 1 {n^{1/n}}$).