<s>[INST] Find the value of $\sin(\tfrac{\pi}3) + \tfrac12\sin(\tfrac{2\pi}3) + \tfrac13\sin(\tfrac{3\pi}3)+\dots$ up to infinity Initially, I thought of doing this by first evaluating the function 
f(x) = cos(x) + cos(2x) + cos(3x) +.....
and then integrating it. However, I cant seem to find a proper integratable function (if that's a word) to replace f(x) with. Am I on the right track, or is the solution on a completely different path? Anyways, I would appreciate solutions of 10+2 level, if it exists.
 [/INST] $$\sum^{\infty}_{j=1}\frac{1}{j}\sin \bigg(\frac{j\pi}{3}\bigg)$$
$$=\frac{\sqrt{3}}{2}\bigg[1+\frac{1}{2}-\frac{1}{4}-\frac{1}{7}+\frac{1}{8}-\frac{1}{10}-\frac{1}{11}+\cdots\bigg]$$
$$=\frac{\sqrt{3}}{2}\int^{1}_{0}\bigg(1+x-x^3-x^4+x^6+x^7-x^9-x^{10}\cdots\bigg)dx=\frac{\sqrt{3}}{2}\int^{1}_{0}\frac{1+x}{1+x^3}dx$$
So Sum is $$\frac{\sqrt{3}}{2}\int^{1}_{0}\frac{1}{1-x+x^2}dx=\frac{\pi}{3}.$$
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