Root Test - Proof

Proof

The proof of the convergence of a series Σa_n is an application of the comparison test. If for all n ≥ N (N some fixed natural number) we have then Since the geometric series converges so does by the comparison test. Absolute convergence in the case of nonpositive a_n can be proven in exactly the same way using

If for infinitely many n, then a_n fails to converge to 0, hence the series is divergent.

Proof of corollary: For a power series Σa_n = Σc_n(z − p)n, we see by the above that the series converges if there exists an N such that for all n ≥ N we have

equivalent to

for all n ≥ N, which implies that in order for the series to converge we must have for all sufficiently large n. This is equivalent to saying

so Now the only other place where convergence is possible is when

(since points > 1 will diverge) and this will not change the radius of convergence since these are just the points lying on the boundary of the interval or disc, so