5.4 Comparison tests  (Page 3/7)

Note that if $a_n\text{/}{b}_n\to 0$ and ${\displaystyle \sum }_{n=1}^{\infty }{b}_n$ diverges, the limit comparison test gives no information. Similarly, if $a_n\text{/}{b}_n\to \infty$ and ${\displaystyle \sum }_{n=1}^{\infty }{b}_n$ converges, the test also provides no information. For example, consider the two series ${\displaystyle \sum _{n=1}^{\infty }1}\text{/}\sqrt{n}$ and ${\displaystyle \sum _{n=1}^{\infty }1}\text{/}{n}^2.$ These series are both p -series with $p=1\text{/}2$ and $p=2,$ respectively. Since $p=1\text{/}2>1,$ the series ${\displaystyle \sum _{n=1}^{\infty }1}\text{/}\sqrt{n}$ diverges. On the other hand, since $p=2<1,$ the series ${\displaystyle \sum _{n=1}^{\infty }1}\text{/}{n}^2$ converges. However, suppose we attempted to apply the limit comparison test, using the convergent $p-\text{series}$ ${\displaystyle \sum _{n=1}^{\infty }1}\text{/}{n}^3$ as our comparison series. First, we see that

Similarly, we see that

Therefore, if $a_n\text{/}{b}_n\to \infty$ when ${\displaystyle \sum _{n=1}^{\infty }{b}_n}$ converges, we do not gain any information on the convergence or divergence of ${\displaystyle \sum _{n=1}^{\infty }{a}_n}.$

Key concepts

• The comparison tests are used to determine convergence or divergence of series with positive terms.
• When using the comparison tests, a series ${\displaystyle \sum }_{n=1}^{\infty }{a}_n$ is often compared to a geometric or p -series.

Use the comparison test to determine whether the following series converge.