# Limit and Convergence

1. Nov 6, 2013

Suppose a_n > 0 and b_n > 0 for all n in natural number (N). Also, lim a_n/b_n = 0 as n goes to infinity. Then the sum of a_n converges if and only if the sum of b_n converges ...both from 1 to infinity.

My approach is that lim a_n/b_n = 0 means that there exists N in natural number (N) for which |a_n/b_n - 0| < 0 for all n >= N. Then 0 < a_n < 0. The sum of a_n from 1 to infinity is 0. So The sum of a_n from 1 to infinity is convergent.

Is this proof that easy or I miss something?

2. Nov 6, 2013