p-Series

Definition

Convergence

Proof

Using the integral test:

Let $f(x) = \frac{1}{x^p}$, then $f(x)$ is continuous, monotonically decreasing, and non-negative on $[1, +\infty)$.

The convergence of the integral $\int_1^{+\infty} \frac{1}{x^p} dx$:

When $p \neq 1$: $\int_1^{+\infty} \frac{1}{x^p} dx = \frac{x^{1-p}}{1-p} \big|_1^{+\infty}$

• When $p > 1$, the integral converges
• When $p < 1$, the integral diverges

When $p = 1$: $\int_1^{+\infty} \frac{1}{x} dx = \ln x \big|_1^{+\infty} = +\infty$

So the integral diverges.

Examples

Example 1: Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$.

Solution: This is a p-series with $p = 2 > 1$

Therefore, the series converges.

Example 2: Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^3}$.

Solution: This is a p-series with $p = 3 > 1$

Therefore, the series converges.

Example 3: Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$.

Solution: This is a p-series with $p = \frac{1}{2} \leq 1$

Therefore, the series diverges.

Exercises

Exercise 1

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^3}$.

Exercise 2

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^4}$.

Exercise 3

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$.

Exercise 4

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^{1.5}}$.

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Summary

Symbols Used in This Article