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It seems that the summation in the original problem should start at $n = 0$. We prove the modified identity by evaluating the integral vis the Euler beta and gamma functions. Indeed, \begin{eqnarray*} \int_0^1\,\frac{2-3x}{1-x^2 +x^3}\,dx & = & \int_0^1\,\frac{2-3x}{1-x^2 (1-x)}\,dx\\ & ...

Here is an elementary way: Let the series be $S$. Appealing to $$\frac{k+1}{k^2 + 2k} = \frac{1}{2}\,\left(\frac{1}{k} + \frac{1}{k+2}\right),$$ we find that \begin{eqnarray*} S & = & \frac{1}{2}\,\left(\sum_{k=1}^\infty\,(-1)^k\frac{1}{k} + \sum_{k=1}^\infty\,(-1)^k\frac{1}{k+2}\right)\\ &a...

Let $a_n$ be a nonnegative and decreasing sequence. Show that $\sum_{n=1}^\infty\,a_n\sin nx$ converges uniformly on $\mathbb{R}$ if and only if $\lim_{n \to \infty}\,na_n = 0$. $(\Rightarrow)$ Assume the series $\sum_{n=1}^\infty\,a_n\sin nx$ converges uniformly on $\mathbb{R}$. Then for every $\ep...