Checking the Convergence of a Series Using the Integral Test

Introduction to Convergence Test: Determining whether a series converges or diverges is a fundamental aspect of calculus and advanced mathematics. One of the primary methods to analyze series is through the Integral Test. This article provides a detailed explanation on how to apply the Integral Test to determine the convergence of a specific series.

Consider the following series:

Displaystyle sum_{n2}^{infty} frac{1}{n ln^n n}

To determine if this series converges or diverges, we can use the Integral Test, which states that if we can evaluate the improper integral:

Displaystyle int_2^{infty} frac{1}{x ln^p x} dx

Applying the Integral Test

Let's proceed with the integral test step-by-step.

Step 1: Define the Integral

Consider the integral:

I int_2^{infty} frac{1}{x ln^p x} dx

Step 2: Substitution

Substitute u ln x, hence du frac{1}{x} dx. This transforms the integral as follows:

I int_{ln 2}^{infty} frac{1}{u^p} du

Step 3: Evaluate the Integral

The integral can be evaluated based on the value of p.

If p 1, the integral becomes:

int_{ln 2}^{infty} frac{1}{u} du lim_{b to infty} [ln u]_{ln 2}^{b} lim_{b to infty} (ln b - ln(ln 2)) infty

Step 4: Evaluate for Other Values of p

For p neq 1:

int_{ln 2}^{infty} frac{1}{u^p} du lim_{b to infty} [frac{u^{1-p}}{1-p}]_{ln 2}^{b}

Check different cases:

If p 1:

lim_{b to infty} [frac{u^{1-p}}{1-p}]_{ln 2}^{b} lim_{b to infty} (frac{b^{1-p}}{1-p} - frac{(ln 2)^{1-p}}{1-p}) infty

If p 1:

lim_{b to infty} [frac{u^{1-p}}{1-p}]_{ln 2}^{b} lim_{b to infty} (frac{b^{1-p}}{1-p} - frac{(ln 2)^{1-p}}{1-p}) frac{(ln 2)^{1-p}}{p-1}

Conclusion

The series converges if and only if the integral converges. From the integral test, we determine:

For p 1, the integral diverges. For p 1, the integral converges. Hence, the series converges precisely when p 1.

Key Takeaways

The Integral Test provides a powerful tool for analyzing the convergence of series. The convergence of the given series depends on the value of p. For p 1, the series diverges; for p 1, the series converges.

Related Keywords

integral test, convergence of series, integral convergence