Question: Let $p_1<p_2<p_3<\cdots$ be all the odd primes.
(1) Show that $$S=\sum_{k=1}^{\infty}\frac{(-1)^{(p_k+1)/2}}{p_k}$$ diverges.
(2) Show that $S$ converges to a real in $(0, \frac{1}{2})$.
(3) Can the terms in the sum be rearranged such that it converges to a real number $T\neq S$?
I have a solution for part (1), but really have no idea how to approach parts (2) and (3). For part (1), notice that the sum is just $$\sum_{k=1}^{\infty}(-1)\left(\frac{-1}{p}\right)\frac{1}{p_k}.$$ This converges since the absolute value of the sum is of the form $$\sum_{p_k}\frac{\chi(p_k)}{p_k},$$ where $\chi$ is a nontrivial character (here the legendre symbol $(-1|p)$), which converges (this is sometimes used in the proof of Dirichlet's theorem, I think).
Any help on (2) and (3) is appreciated.