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:

  1. If $\displaystyle{\sum_{n=1}^\infty b_n}$ converges and $a_n \le b_n$ for all $n$, then $\displaystyle{\sum_{n=1}^\infty a_n}$ converges.

  2. If $\displaystyle{\sum_{n=1}^\infty b_n}$ diverges and $a_n \ge b_n$ for all $n$, then $\displaystyle{\sum_{n=1}^\infty a_n}$ diverges.
In fact, 1. will work if $a_n\le b_n$ for all $n$ larger than some finite positive $N$, and similarly for 2.


Example 1: 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 $\displaystyle \sum_{n=1}^\infty\left(\frac{2}{3}\right)^n$ is a convergent geometric series, with $r=\frac23<1$.

The video explains the test, and looks at an example.


Example 2:  Test the series $\displaystyle\sum_{k=1}^\infty\frac{\ln k}{k}$ for convergence or divergence.

DO:  Do you think this series converges?  Try to figure out what to compare this series to before reading the solution


Solution 2:  $\displaystyle\frac{\ln k}{k}\ge\frac1k$ when $k>2$, and the harmonic series with terms $\frac1k$ diverges, so our series diverges.

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Example 3:  Test the series $\displaystyle\sum_{n=1}^\infty\frac{1}{5n+10}$ for convergence or divergence.  DO:  Try this before reading further.


Solution 3:  The terms look much like the harmonic series, and when we compare terms, we see that $\displaystyle\frac{1}{5n+10}\le\frac1n$.  But the harmonic series diverges.  Our terms are smaller than those of a divergent series, so we know nothing.  Let's compare to $\displaystyle\frac1{n^2}$.  The series $\displaystyle\sum\frac{1}{n^2}$ is a convergent $p$-series, with $p=2$.  But when we compare terms, we get $\displaystyle\frac{1}{5n+10}\ge\frac1{n^2}$ as long as $n\ge7$, so our terms are larger than those of a convergent series, and this comparison also tells us nothing.  We will use the limit comparison test (coming up) to test this series.