<s>[INST] Sum of the reciprocals of the odds (alternating) I've been trying to work out what the following infinite series converges to:
$$\frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11}...$$
But evaluating something like this is still new to me, and I haven't found any answers online (perhaps I'm not googling well enough?).So my question is, what value does it converge to and how is it derived?
 [/INST] As mentioned by @KentaS, you can proceed as follows (for $|x| < 1$):
\begin{align*}
\frac{1}{1 - x} = 1 + x + x^{2} + x^{3} & \Rightarrow \frac{1}{1 + x^{2}} = 1 - x^{2} + x^{4} - x^{6} + \ldots\\\\
& \Rightarrow \arctan(x) = x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \ldots\\\\
& \Rightarrow \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \ldots
\end{align*}
It is worth mentioning that, even though the geometric series does not converge for $x = 1$, the expression deduced for $\arctan$ also holds true when $x = 1$. To conclude so, it suffices to apply the Leibniz test as suggested by @Hussain-Alqatari in the comments.
Hopefully this helps!
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