Question

Use the Integral Test to determine the convergence of the given series. $$\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{2}}$$

Short answer

The series converges as the integral evaluates to a finite value.

Step-by-step solution

Identify the Integral

To use the Integral Test, we first need to consider the function \( f(x) = \frac{1}{x(\ln x)^2} \) as the continuous counterpart of the series \( \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{2}} \). We will evaluate the integral from 2 to infinity: \[ \int_{2}^{\infty} \frac{1}{x(\ln x)^2} \, dx \].

Check Conditions for the Integral Test

For the Integral Test to be applicable, the function \( f(x) = \frac{1}{x (\ln x)^2} \) must be continuous, positive, and decreasing for \( x \geq 2 \). Since \( x > 1 \) ensures \( \ln x > 0 \), and the derivative \( f'(x) < 0 \) for \( x \geq 2 \), the function meets these conditions.

Determine the Antiderivative

We will use substitution to find the antiderivative. Let \( u = \ln x \), then \( du = \frac{1}{x} dx \). The integral becomes: \[ \int \frac{1}{x (\ln x)^2} \, dx = \int \frac{1}{u^2} \, du \].

Integrate the Function

The integral \( \int \frac{1}{u^2} \, du \) is \( -\frac{1}{u} + C \). Substituting back \( u = \ln x \), the integral becomes: \[ -\frac{1}{\ln x} + C \].

Evaluate the Improper Integral

Now we evaluate the improper integral: \[ \lim_{b \to \infty} \int_{2}^{b} \frac{1}{x (\ln x)^2} \, dx = \lim_{b \to \infty} \left[ -\frac{1}{\ln x} \right]_{2}^{b} = \lim_{b \to \infty} \left( -\frac{1}{\ln b} + \frac{1}{\ln 2} \right) \]. As \( b \to \infty \), \( \frac{1}{\ln b} \to 0 \). Therefore, the integral evaluates to \( \frac{1}{\ln 2} \).

Conclusion on Convergence

Since the improper integral evaluates to a finite value, the original series \( \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{2}} \) converges by the Integral Test.

Key concepts

Convergence of Series

The Integral Test is a fundamental technique for determining whether a series converges or diverges. In general, series convergence refers to whether the sum of all the terms in the series approaches a finite value. When using the Integral Test, if we have a series \( \sum_{n=2}^{\infty} a_n \), we can relate it to an integral. This involves comparing the sum of the series to the integral of a function \( f(x) \), where the function must preserve the properties of the series.

• The function \( f(x) \) must be continuous on the interval \([2, \infty)\).
• It should be positive, ensuring that each term adds uniquely to the total sum and does not detract.
• The function must be decreasing, showing that each subsequent term is smaller than the preceding one.

If the integral \( \int_{2}^{\infty} f(x) \, dx \) converges to a finite value, then the original series \( \sum_{n=2}^{\infty} a_n \) is also convergent.

Antiderivative

Finding the antiderivative, also known as the indefinite integral, is a vital step in evaluating integrals. The antiderivative is a function whose derivative equals the original function, essentially reversing differentiation. For example, when evaluating \[ \int \frac{1}{u^2} \, du \] we find the antiderivative to be \[ -\frac{1}{u} + C \]where \( C \) is the constant of integration. This process enables us to express the indefinite integral in terms of more elementary functions. To effectively utilize this method, understanding how different functions can be integrated and the appropriate techniques for integration, such as substitution, plays a significant role. This knowledge is crucial for transforming complex integrals into simpler, solvable forms.

Improper Integral

Improper integrals are integrals with one or both limits of integration extending to infinity, or where the integrand becomes infinite within the limits of integration. Evaluating improper integrals requires taking the limit of the integral as one of the endpoints approaches infinity. Consider an example: Suppose we have an integral \[ \int_{2}^{\infty} \frac{1}{x(\ln x)^2} \, dx \].To solve this, we evaluate \[ \lim_{b \to \infty} \int_{2}^{b} \frac{1}{x(\ln x)^2} \, dx \].The limit process captures the behavior of the function as it extends beyond any finite bound. If the limit yields a finite result, then the improper integral is said to converge. Otherwise, it diverges. This method helps ensure that we quantify the "area" under a curve, even when extending indefinitely.

Substitution Method in Integration

The substitution method in integration, frequently referred to as "\( u \)-substitution," simplifies complex integrals by changing variables. By doing so, it transforms the original integral into a potentially easier form to evaluate.To illustrate, for the function \( \int \frac{1}{x (\ln x)^2} \, dx \),we employ the substitution \( u = \ln x \),which changes the differential: \( du = \frac{1}{x} \, dx \).This transformation simplifies our integral to \( \int \frac{1}{u^2} \, du \),which has a straightforward antiderivative \( -\frac{1}{u} + C \).Ultimately, once the "simplified" integral is evaluated, we revert \( u \) back to \( x \) to express the antiderivative in terms of the original variable. Substitution is a powerful tool for transforming intricate integrals into forms that can be tackled with basic integration rules, making it invaluable for students and professionals alike.