Which series should we compare this to?

What is the value of the limit in the LCT?

Describe the behavior of the series

We will use the LCT to determine if a series converges or diverges.

We have seen that the Direct Comparison Test can be inconclusive if the comparison goes in the wrong direction. In these cases, the Limit Comparison Test (LCT) can be used instead. As its name suggests, the LCT involves computing a limit. More precisely, it involves computing the limit of the ratio of a given series with another series whose behavior is known. The idea is that if the limit of the ratio of these two series is a positive number, , then the two series will have the same behavior, as one of them is essentially a multiple of the other. We state this in the following theorem.

Limit Comparison Test Suppose for all . If where then the two infinite series have
the same behavior, i.e., they either both converge or both diverge.

In practical applications of the LCT, the given series is and the series we choose to
compare it with is .

The LCT is inconclusive if the limit DNE, but if the limit is zero or infinity, then there is a partial conclusion as mentioned in the following remark.

example 1 Determine if the series converges or diverges.(We saw this series in the
section on the DCT and were not able to make a conclusion).

We will use the LCT with the convergent p-series: . We have: using the ratio of the leading coefficients. We now make our conclusion using the value . Since and the series converges, the series also converges by the LCT.

We will use the LCT with the convergent p-series: . We have: using the ratio of the leading coefficients. We now make our conclusion using the value . Since and the series converges, the series also converges by the LCT.

(problem 1) Determine if the series converges or diverges:

Which series should we compare this to?

Which series should we compare this to?

What is the value of the limit in the LCT?

Describe the behavior of the series

Converges by LCT Diverges by LCT No
Conclusion from LCT

example 2 Determine if the series converges or diverges.

We will use the LCT with the convergent geometric series: . We have:

We will use the LCT with the convergent geometric series: . We have:

using L’Hopital’s rule on each of the last two limits. We now make our conclusion using the value . Since and the series converges, the series also converges by the LCT.

(problem 2) Determine if the series converges or diverges:

Which series should we compare this to?

Which series should we compare this to?

What is the value of the limit in the LCT?

Describe the behavior of the series

Converges by LCT Diverges by LCT No
Conclusion from LCT

example 3 Determine if the series converges or diverges.

We will use the LCT with the divergent harmonic series: . We have: using the ratio of the leading coefficients. We now make our conclusion using the value . Since and the series diverges, the series also diverges by the LCT.

We will use the LCT with the divergent harmonic series: . We have: using the ratio of the leading coefficients. We now make our conclusion using the value . Since and the series diverges, the series also diverges by the LCT.

(problem 3) Determine if the series converges or diverges:

Which series should we compare this to?

Which series should we compare this to?

What is the value of the limit in the LCT?

Describe the behavior of the series

Converges by LCT Diverges by LCT No
Conclusion from LCT

example 4 Determine if the series converges or diverges.

We will use the LCT with the divergent harmonic series: . We have: We now make our conclusion using the value . Since and the series diverges, the series also diverges by the LCT.

We will use the LCT with the divergent harmonic series: . We have: We now make our conclusion using the value . Since and the series diverges, the series also diverges by the LCT.

(problem 4) Determine if the series converges or diverges:

Which series should we compare this to?

Which series should we compare this to?

What is the value of the limit in the LCT?

Describe the behavior of the series

Converges by LCT Diverges by LCT No
Conclusion from LCT

Since the limits do not exist (due to oscillation), the LCT is generally not effective in
problems involving and . Try the DCT instead.

Here is a detailed, lecture style video on the Limit Comparison Test:

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(problem 5) Determine if the series converges or diverges:

Which series should we compare this to?

Which series should we compare this to?

What is the value of the limit in the LCT?

Describe the behavior of the series

Converges by LCT Diverges by LCT No
Conclusion from LCT

(problem 6) Determine if the series converges or diverges:

Which series should we compare this to?

Which series should we compare this to?

What is the value of the limit in the LCT?

Describe the behavior of the series

Converges by LCT Diverges by LCT No
Conclusion from LCT