Was reading Thomas Calculus and found this loophole which I can't fill up could someone give an outline of this prove/explain? Much appreciated!
Prove that SEQUENCE (not series) of terms of a power series must diverge outside the radius of convergence of a power series, except possibly ON the boundary point of interval of convergence
eg. consider the series sum(x^n/n) radius of convergence: [-1,1), sequence nevertheless converges at x=1 (harmonic series), but diverges elsewhere, we need to show that other series like as the harmonic series does not exist elsewhere outside the radius of convergence
What I tried was considering power series at c and c+epsilon, but the binomial expansion of the latter became quite complicated and I do not know how to use the fact that the series diverges for all numbers greater than c