Past exam of the mathematics course of the University of Cambridge / 2004 / ia / Paper 1 / 12F / i / Solution

Rearranging the assumed positive-term ratio inequality givesInduction therefore yields $a_n/b_n\le a_1/b_1$ for every $n\ge 1$. In particularThe positive partial sums of the first series are nondecreasing and bounded above, so the bounded monotone sequence theorem proves that they converge. This is the ratio comparison test, and establishes convergence of $\sum a_n$.