Problem : What is the radius of convergence of xn? of (- 1)nxn?Since the radius of convergence only has to do with absolute convergence, the answer to the two parts will be the same. Since xn is a geometric series, it converges whenever x≤1, but does not converge when x = 1. Therefore, r = 1 is its radius of convergence.
Problem : Using the fact that
|ex = 1 + x + + + ... + + ...|
whenever the right hand series makes sense, find a series that converges to e. We have seen that the radius of convergence of the right-hand series is ∞. We can therefore substitute x = 1 into this equation, and
|e = 1 + 1 + + + ... + + ...|