TITLE: Convergence of $\lim\limits_{n\to\infty}\sum\limits_{k=1}^{n}\binom{n}{k}\frac{\sin k}{k}$
QUESTION [14 upvotes]: Is this series convergent? I met it when I was studying some fractals.
$$\lim_{n\to\infty}\sum_{k=1}^{n}\binom{n}{k}\frac{\sin k}{k}$$

REPLY [2 votes]: Before taking the limit, the sum has an expression form in terms of hypergeometric functions:
$$
\sum _{k=1}^n \binom{n}{k}\frac{\sin k }{k} = 
-\frac{1}{2} i e^{-i} n \left(e^{2 i} \, _3F_2\left(1,1,1-n;2,2;-e^i\right)-\, _3F_2\left(1,1,1-n;2,2;-e^{-i}\right)\right)
$$
Now you just need to study the asymptotics of such functions, namely, do they decay fast enough as the thrid argument goes to $-\infty$.  I think you could find these properties in the usual references about special functions (DLMF or Abramowitz and Stegun), and that would suffice as a proof.  I can certainly say that they grow unboundedly as the third argument becomes increasingly negative, but you'd need some asymptotic bounds to finish off a proof.