Alternating Series

Theory

An alternating series has the form

$$\sum (-1)^{n+1} a_n \quad \text{or} \quad \sum (-1)^n a_n,$$

where $a_n \ge 0$ for all $n$.

According to the Alternating Series Test, the series converges if:

• $a_n$ is decreasing, and
• $\lim_{n\to\infty} a_n = 0$.

Exercises

Exercise 1

Question

Eg. a. Determine if the following alternating series converges or diverges.

$$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$$

Show solution ↓Hide solution ↑

Solution

Let $a_n=\frac{1}{n}$.

a_{n+1}=\frac{1}{n+1}<\frac{1}{n}, \quad \text{and} \quad \lim_{n\to\infty} a_n=0.

Conclusion: The series converges (conditionally).

Exercise 2

Question

Eg. b. Determine if the following alternating series converges or diverges.

$$\sum_{n=1}^\infty (-1)^{n+1}\frac{2^n}{n^2}$$

Show solution ↓Hide solution ↑

Solution

Let $a_n=\frac{2^n}{n^2}$. Since $2^n$ grows faster than $n^2$,

$$\lim_{n\to\infty} a_n=\infty \neq 0.$$

Conclusion: The series diverges.