In looking at a paper online I came across the following proposition:
$1 - \frac{1}{9} - \frac{1}{15}- \frac{1}{21}-\cdots = 0$
After wasting a lot of time, I rewrote it,
$1 -\left(\frac{1}{9} + \frac{1}{15}+\cdots\right)= 1 - \left(\frac{1}{3}\sum_{k = 2}^\infty\frac{1}{p_k}\right) \rightarrow -\infty. $
So it seems to me that the r.h.s. diverges. Is this correct?
Edit: in case someone doesn't see Sasha's question and my response here are more terms in the sequence. The denominators are $3 p_k, p_k$ being the $k$th prime. $1- 1/9 - 1/15 - 1/21 - 1/33 - 1/39 - 1/51-\cdots$