Sequences and Series, Root Test, Ratio Test

Root Test, Ratio Test

To test the convergence of the complex series s, assume the limit of s_n^1/n = w (root test), or the limit of s_n+1/s_n = w (ratio test). In either case, if |w| < 1 the series converges, and if |w| > 1 the series diverges.

For the moment, replace s_j with |s_j|.

Assume w exceeds 1, and let v be any number between 1 and w. The root test tells us that beyond some n, s_j exceeds v^j. The norms of the terms of s are unbounded, hence the terms of s do not approach 0, and s diverges. Similarly, the ratio test tells us that the tail of the series exceeds s_n×v^j. (Use induction to prove this.) Once again the norms are unbounded and s diverges.

If w < 1, let v be a number between w and 1. This time the root test or the ratio test tells us that the tail of s is dominated by a geometric series that converges. In fact it converges absolutely, hence the norms of s converge absolutely, hence s converges absolutely.