Wolfram alpha is saying that the series of $\sum_k\sin(2 k \arctan(k^2))$ does not converge:
http://www.wolframalpha.com/input/?i=sum+sin%282+k+atan%28k%5E2%29%29
instead it converges! Seems that mathematica is only dealing with limits of functions not with limit of sequences.
Another simpler example is $\sum_k \sin(2k \pi + 1/k^2)$:
http://www.wolframalpha.com/input/?i=sum+sin%282k+pi+%2B+1%2Fk%5E2%29
E.
