Does the Integral Test $\sum_{n=1}^{\infty}\frac{\ln n}{n^p}$ Converge?

Thread starter nameVoid
Start date Apr 26, 2010
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Integral Integral test Test

In summary, the Integral Test is a mathematical tool used to determine the convergence or divergence of an infinite series by comparing it to a corresponding improper integral. It works by comparing the terms of the series to the integral and can be used for series with positive terms that may not have a clear pattern. The Integral Test differs from the Comparison Test, which compares the series to another known series. However, the Integral Test has limitations as it can only be applied to series with positive terms and requires a continuous, positive, and decreasing function on the interval of integration that can be solved using known methods.

Apr 26, 2010

nameVoid

[tex]
\sum_{n=1}^{\infty}\frac{lnn}{n^P}
[/tex]
[tex]
\int_{1}^{\infty}lnx(x^{-P})dx
[/tex]

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Apr 26, 2010

lanedance

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Related to Does the Integral Test $\sum_{n=1}^{\infty}\frac{\ln n}{n^p}$ Converge?

1. What is the Integral Test?

The Integral Test is a mathematical tool used to determine the convergence or divergence of an infinite series. It is based on the comparison of the series to a corresponding improper integral.

2. How does the Integral Test work?

The Integral Test works by comparing the terms of an infinite series to the corresponding integral. If the integral converges, then the series also converges. If the integral diverges, then the series also diverges.

3. When should the Integral Test be used?

The Integral Test is most useful for determining the convergence or divergence of series with positive terms that may not have a clear pattern or ratio between terms. It is also helpful for series with terms that approach zero as n approaches infinity.

4. What is the difference between the Integral Test and the Comparison Test?

The Integral Test and the Comparison Test are both used to determine the convergence or divergence of an infinite series. However, the Integral Test compares the series to a corresponding integral, while the Comparison Test compares the series to another known series.

5. Are there any limitations to the Integral Test?

Yes, the Integral Test can only be applied to series with positive terms. It also requires the function to be continuous, positive, and decreasing on the interval of integration. Additionally, the integral must be solvable using known methods.