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how to find that summation by Mathematica Wolfram

\(\displaystyle \sum_{x\in A} \sin(\pi x) \)

\(\displaystyle A={{1,\frac{1}{2},\frac{1}{3},\frac{1}{4},.........}}\)
 

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how to find that summation by Mathematica Wolfram

\(\displaystyle \sum_{x\in A} \sin(\pi x) \)

\(\displaystyle A={{1,\frac{1}{2},\frac{1}{3},\frac{1}{4},.........}}\)
Input: Sum[Sin[Pi/n], {n, 1, Infinity}].

Message: Sum::div: Sum does not converge.
 

CaptainBlack

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how to find that summation by Mathematica Wolfram

\(\displaystyle \sum_{x\in A} \sin(\pi x) \)

\(\displaystyle A={{1,\frac{1}{2},\frac{1}{3},\frac{1}{4},.........}}\)

Since for small \(\displaystyle u\) we have \(\displaystyle \sin(u)\approx u\) this is obviously not convergent as the tail of the sum behaves like the harmonic series which is known to be divergent.
.

CB