Give an example, with proof, of a power series that converges exactly on the interval \([3,5)\).
Solution
We want some power series centered at 4 with convergence radius 1, so something of the form \(\sum_n f(n)(x-4)^n\) with \(L = \lim \frac{a_{n+1}}{a_n} = 1\).
Then, by plugging in \(5\) and \(3\), we find that we want \(\sum (-1)^n f(n)\) to converge but \(\sum f(n)\) to diverge. The canonical example of this is the harmonic series with \(f(n) = \frac 1 n\), so we can just take \[f(x) = \sum_n \frac{(x-4)^n}{n}. \qed\]