Suppose that \(\sum a_n\) and \(\sum b_n\) are series with positive terms, and suppose:
\[ \lim_{n \to \infty} \frac{a_n}{b_n} = c \]
where \(c\) is a finite positive number. Let there be integers \(m\) and \(M\) such that \(m \lt c \lt M\). Since \(a_n / b_n\) is close to \(c\) for large \(n\), there is an integer \(N\) such that when \(n \gt N\):
\[ m \lt \frac{a_n}{b_n} \lt M \]
So:
\[ mb_n \lt a_n \lt Mb_n \]
If \(\sum b_n\) converges, then so does \(\sum Mb_n\). This means by the comparison test, \(\sum a_n\) also converges. If \(\sum b_n\) diverges, then so does \(\sum mb_n\). This means by the comparison test, \(\sum a_n\) also diverges.