An archived instance of discourse for discussion in undergraduate Real and Numerical Analysis.

A fun sum


This is an expansion of problem 6.6.1 from the text.

Starting from the geometric series, derive a power series representation for $\arctan(x)$ plug in $x=1$ to obtain a nice series expression for $\pi/4$. Now prove that your series actually converges to $\pi/4$.

Note: Our original series (the geometric series) did not converge at $x=1$. The new series does converge conditionally at $x=1$, but some care must be taken taken to ensure that it converges to the correct thing. Abel's theorem on convergence will likely help.