EDB — 1KJ

Exercises

1. [1KJ]Find an example of a series \(f(t)=∑_{k=0}^∞ a_ k t^ k\) with \(a_ k∈ℝ\) and with radius of convergence \(r\) positive and finite, such that the limit \(\lim _{t→ r-} f(t)\) exists and is finite, but the series does not converge in \(t=r\).

   Solution 1

   [1KK]↺↻

   Note that (by Abel’s lemma) if the series converges in \(t=r\) then the limit \(\lim _{t→ r-} f(t)\) exists and \(\lim _{t→ r-} f(t)=f(r)\).