Question

Use the integral test to find whether the following series converge or diverge. Hint and warning: Do not use lower limits on your integrals (see Problem 16 ). $$\sum_{n=2}^{\infty} \frac{1}{n \ln n}$$

Short answer

The series \(\textstyle\boldsymbol{\sum_{n=2}^{\infty} \frac{1}{n \ln n}}\) diverges.

Step-by-step solution

- Define the function

The given series is \(\textstyle\boldsymbol{\begin{array}{l}\sum_{n=2}^{\infty} \frac{1}{n \ln n}\end{array}}\) . Define the function \(\textstyle\boldsymbol{f(x) = \frac{1}{x \ln x}}\) , which is a continuous, positive, and decreasing function for \(\textstyle\boldsymbol{x \geq 2}\) .

- Set up the improper integral

To use the integral test, set up the improper integral \(\textstyle\boldsymbol{\int_{2}^{\infty} \frac{1}{x \ln x} \, dx}\) and evaluate its convergence or divergence.

- Use substitution

Use the substitution \(\textstyle\boldsymbol{u = \ln x \implies du = \frac{1}{x} dx}\). This transforms the integral into \(\textstyle\boldsymbol{\int_{\ln 2}^{\infty} \frac{1}{u} \, du}\) .

- Evaluate the integral

Integrate \(\textstyle\boldsymbol{\int_{\ln 2}^{\infty} \frac{1}{u} du = \left.[\ln|u|\right]_{\ln 2}^{\infty} = \lim_{{b \to \infty}} [\ln(b) - \ln(\ln 2)]}\). As \(\textstyle\boldsymbol{b \to \infty}\), \(\textstyle\boldsymbol{\ln(b) \to \infty}\), hence the integral diverges.

- Conclude using the integral test

Since the integral \(\textstyle\boldsymbol{\int_{2}^{\infty} \frac{1}{x \ln x} dx}\) diverges, by the integral test, the series \(\textstyle\boldsymbol{\sum_{n=2}^{\infty} \frac{1}{n \ln n}}\) also diverges.

Key concepts

Improper Integrals

Improper integrals are used to evaluate integrals with infinite limits or integrands that approach infinity within the interval of integration. Typical notation involves integrals like \(\textstyle \boldsymbol{\begin{array}{l} \int_{a}^{\boldsymbol{\bigsurd}} f(x) \, dx \end{array}} \) or \(\textstyle \boldsymbol{\begin{array}{l} \int_{a}^{b} f(x) \, dx \end{array}} \) where \(\textstyle \boldsymbol{f(x)}\) has a vertical asymptote at some point within \(\textstyle \boldsymbol{(a,b)}\).

Since improper integrals deal with limits, we often use limits to evaluate them.
For example, to evaluate \(\textstyle \boldsymbol{\begin{array}{l} \int_{a}^{\bigsurd} f(x) \, dx \end{array}} \), we consider \(\textstyle \boldsymbol{\begin{array}{l} \int_{a}^{b} f(x) \, dx \end{array}} \) and then take \(\textstyle \boldsymbol{\begin{array}{l} \lim_{{b \to \boldsymbol{\bigsurd}}}} \end{array}} \).

To decide if an improper integral converges or diverges, follow these steps:

• Set up the integral with correct limits.
• Apply necessary substitutions, if needed, to simplify the expression.
• Evaluate the limit of the integral as it approaches the upper or lower limits.

Series Convergence and Divergence

Determining whether a series converges or diverges is crucial in mathematical analysis. Convergence means that the sum of the series approaches a finite value, while divergence means that the sum increases without bound.

To check convergence or divergence, various tests can be applied, such as:

• The Integral Test
• The Comparison Test
• The Ratio Test
• The Root Test

Convergence or divergence helps us understand the behavior of functions, especially in computing series approximations.

For our problem, we used the integral test which involves comparing the series to an improper integral. By evaluating the improper integral, we determine the convergence or divergence of the series.

Substitution Method

Substitution method, or u-substitution, is a technique used to simplify the integration process by changing variables. It is particularly useful when dealing with complicated integrals or when dealing with transformations in improper integrals.

In the context of our exercise, we used the substitution \(\textstyle \boldsymbol{u = \ln x}\). This allowed us to transform the integral \(\textstyle \boldsymbol{\int_{2}^{\bigsurd} \frac{1}{x \ln x} dx}\) into a simpler form, making it easier to evaluate:

• Substitute \(\textstyle \boldsymbol{u}\) for \(\textstyle \boldsymbol{\ln x}\).
• The differential \(\textstyle \boldsymbol{du = \frac{1}{x} dx}\).
• The integral then becomes \(\textstyle \boldsymbol{\begin{array}{l}\int_{\boldsymbol{\bigsurd}} \frac{1}{u} du} \end{array}}\) with the limits changing accordingly.

After substitution, the integral's evaluation simplifies significantly, helping in determining the series behavior.

Integral Test

The Integral Test is a method used to determine the convergence or divergence of an infinite series. The test compares the series to an improper integral.

To apply the Integral Test, follow these steps:

• Define a function \(\textstyle \boldsymbol{f(x)}\) that is positive, continuous, and decreasing for \(\textstyle \boldsymbol{x}\) greater than or equal to some value.


• Set up the improper integral corresponding to the series: \(\textstyle \boldsymbol{\begin{array}{l} \int_{a}^{\bigsurd} f(x) \, dx \end{array}} \).

If the integral converges, then the series \(\textstyle \boldsymbol{\begin{array}{l} \sum_{n=a}^{\bigsurd} f(n) \end{array}} \) converges.
If the integral diverges, then the series diverges as well.

In our example, we defined \(\textstyle \boldsymbol{f(x) = \frac{1}{x \ln x}}\), set up the improper integral, and determined its divergence. This led us to conclude the series diverges.