<s>[INST] Find the sum of the series: $\frac{1}{1*2} - \frac{1}{3*2^3} + \frac{1}{5*2^5} - \frac{1}{7*2^7}+\dots$? $$\frac{1}{1*2} - \frac{1}{3*2^3} + \frac{1}{5*2^5} - \frac{1}{7*2^7}+\dots$$ 
I made a series to get $$\sum_{n=0}^\inf \frac{(-1)^n}{(1+2n)*2^{1+2n}}$$ but what series can it manipulate and simplify to?
 [/INST] The geometric series gives $$\frac{1}{1+x^2} = \sum_{n=0}^{\infty} (-1)^n x^{2n}, \,\,\,\,\,\,\,\, \text{when } \lvert x \rvert < 1.$$ Then $$\frac{1}{1+x^2} - 1 = \sum_{n=1}^{\infty} (-1)^n x^{2n},$$ for such $x$. Integrating from $x=0$ to $x=1/2$ (and integrating term-by-term in the sum, which is justifies by uniform convergence of the sum) gives $$\arctan\left( \tfrac 1 2\right) - \tfrac{1}{2} = \sum^\infty_{n=1} \frac{(-1)^n \left( \tfrac{1}{2}\right)^{2n+1}}{2n+1} = \sum^\infty_{n=1} \frac{(-1)^n}{(2n+1)2^{2n+1}}.$$
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