Lebl-Basic-Analysis-I · Problem 2.5.11

Sequences and Series

Mathematics · limit-comparison-test

Problem

Prove the limit comparison test. That is, prove that if

a_n > 0

and

b_n > 0

for all

n

, and

0 < \lim_{n\to\infty} \frac{a_n}{b_n} < \infty,

then either

\sum_{n=1}^{\infty} a_n

and

\sum_{n=1}^{\infty} b_n

both converge or both diverge.