The Basic Comparison Test

Theorem: If $\displaystyle{\sum_{n=1}^\infty a_n}$ and $\displaystyle{\sum_{n=1}^\infty b_n}$ are series with non-negative terms, then:

  • If $\displaystyle{\sum_{n=1}^\infty b_n}$ converges and each $a_n \le b_n$, then $\displaystyle{\sum_{n=1}^\infty a_n}$ converges.
  • If $\displaystyle{\sum_{n=1}^\infty b_n}$ diverges and each $a_n \ge b_n$, then $\displaystyle{\sum_{n=1}^\infty a_n}$ diverges.


Example: The series $\displaystyle \sum_{n=1}^\infty\frac{2^n}{3^n+1}$ converges, since $$ \frac{2^n}{3^n+1}\le \frac{2^n}{3^n} $$ and we know that the geometric series $\displaystyle \sum_{n=1}^\infty\left(\frac{2}{3}\right)^n$ converges, as $r=\frac23<1$.