We’re working on power series in Calc. Every year I love setting them up with the following series to find the convergence/divergence of:
They work on whiteboards, wander their way through the tough algebra and to the answer (converges everywhere), and are (generally) very happy to have made it through a tricky problem unscathed. Inevitibly, a student asks what the power series looks like. YES.
It all leads to an amazing conclusion. That mess of a series with alternating terms, odd polynomials, and factorials leads to …… (check the graph to see!)!
And then it also leads to my favorite desmos graph (ever??):