Limit Comparison Test

Learning Outcomes

  • Use the limit comparison test to determine convergence of a series

The comparison test works nicely if we can find a comparable series satisfying the hypothesis of the test. However, sometimes finding an appropriate series can be difficult. Consider the series

[latex]\displaystyle\sum _{n=2}^{\infty }\frac{1}{{n}^{2}-1}[/latex].


It is natural to compare this series with the convergent series

[latex]\displaystyle\sum _{n=2}^{\infty }\frac{1}{{n}^{2}}[/latex].


However, this series does not satisfy the hypothesis necessary to use the comparison test because



for all integers [latex]n\ge 2[/latex]. Although we could look for a different series with which to compare [latex]\displaystyle\sum _{n=2}^{\infty }\frac{1}{\left({n}^{2}-1\right)}[/latex], instead we show how we can use the limit comparison test to compare

[latex]\displaystyle\sum _{n=2}^{\infty }\frac{1}{{n}^{2}-1}\text{ and }\displaystyle\sum _{n=2}^{\infty }\frac{1}{{n}^{2}}[/latex].


Let us examine the idea behind the limit comparison test. Consider two series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex]. with positive terms [latex]{a}_{n}\text{and}{b}_{n}[/latex] and evaluate

[latex]\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}[/latex].



[latex]\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}=L\ne 0[/latex],


then, for [latex]n[/latex] sufficiently large, [latex]{a}_{n}\approx L{b}_{n}[/latex]. Therefore, either both series converge or both series diverge. For the series [latex]\displaystyle\sum _{n=2}^{\infty }\frac{1}{\left({n}^{2}-1\right)}[/latex] and [latex]\displaystyle\sum _{n=2}^{\infty }\frac{1}{{n}^{2}}[/latex], we see that

[latex]\underset{n\to \infty }{\text{lim}}\frac{\frac{1}{\left({n}^{2}-1\right)}}{\frac{1}{{n}^{2}}}=\underset{n\to \infty }{\text{lim}}\frac{{n}^{2}}{{n}^{2}-1}=1[/latex].


Since [latex]\displaystyle\sum _{n=2}^{\infty }\frac{1}{{n}^{2}}[/latex] converges, we conclude that

[latex]\displaystyle\sum _{n=2}^{\infty }\frac{1}{{n}^{2}-1}[/latex]



The limit comparison test can be used in two other cases. Suppose

[latex]\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}=0[/latex].


In this case, [latex]\left\{\frac{{a}_{n}}{{b}_{n}}\right\}[/latex] is a bounded sequence. As a result, there exists a constant [latex]M[/latex] such that [latex]{a}_{n}\le M{b}_{n}[/latex]. Therefore, if [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] converges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges. On the other hand, suppose

[latex]\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}=\infty [/latex].


In this case,[latex]\left\{\frac{{a}_{n}}{{b}_{n}}\right\}[/latex] is an unbounded sequence. Therefore, for every constant [latex]M[/latex] there exists an integer [latex]N[/latex] such that [latex]{a}_{n}\ge M{b}_{n}[/latex] for all [latex]n\ge N[/latex]. Therefore, if [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] diverges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] diverges as well.

Theorem: Limit Comparison Test

Let [latex]{a}_{n},{b}_{n}\ge 0[/latex] for all [latex]n\ge 1[/latex].

  1. If [latex]\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}=L\ne 0[/latex], then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] both converge or both diverge.
  2. If [latex]\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}=0[/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] converges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges.
  3. If [latex]\underset{n\to \infty }{\text{lim}}\frac{{a}_{n}}{{b}_{n}}=\infty [/latex] and [latex]\displaystyle\sum _{n=1}^{\infty }{b}_{n}[/latex] diverges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] diverges.

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

[latex]\frac{\frac{1}{\sqrt{n}}}{\frac{1}{{n}^{3}}}=\frac{{n}^{3}}{\sqrt{n}}={n}^{\frac{5}{2}}\to \infty \text{as}n\to \infty [/latex].


Similarly, we see that

[latex]\frac{\frac{1}{{n}^{2}}}{\frac{1}{{n}^{3}}}=n\to \infty \text{as}n\to \infty [/latex].


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

Example: Using the Limit Comparison Test

For each of the following series, use the limit comparison test to determine whether the series converges or diverges. If the test does not apply, say so.

  1. [latex]\displaystyle\sum _{n=1}^{\infty }\frac{1}{\sqrt{n}+1}[/latex]
  2. [latex]\displaystyle\sum _{n=1}^{\infty }\frac{{2}^{n}+1}{{3}^{n}}[/latex]
  3. [latex]\displaystyle\sum _{n=1}^{\infty }\frac{\text{ln}\left(n\right)}{{n}^{2}}[/latex]

try it

Use the limit comparison test to determine whether the series [latex]\displaystyle\sum _{n=1}^{\infty }\frac{{5}^{n}}{{3}^{n}+2}[/latex] converges or diverges.

Watch the following video to see the worked solution to the above Try It.

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Try It