Infinite series

Definition

An infinite series is the sum of the terms of an infinite sequence. It can converge to a finite value or diverge to infinity or negative infinity.

5 Must Know Facts For Your Next Test

1. The convergence or divergence of an infinite series can often be determined using tests such as the Ratio Test, Root Test, and Integral Test.
2. A geometric series converges if its common ratio $|r| < 1$; otherwise, it diverges.
3. The harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges, even though its terms approach zero.
4. If the terms of a series do not approach zero, $\lim_{{n \to \infty}} a_n \neq 0$, then the series diverges.
5. Power series have a radius of convergence within which they converge absolutely.

Review Questions

• What test would you use to determine the convergence of the series $\sum_{n=1}^{\infty} \frac{3^n}{n!}$?
• Does the series $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ converge or diverge?
• Explain why $\sum_{n=0}^{\infty} ar^n$ converges when $|r| < 1$.

Related terms

Convergence:

When the sum of an infinite series approaches a specific finite value as more terms are added.

Divergence:

When an infinite series does not approach any finite limit as more terms are added.

Power Series:

An infinite series in the form $\sum_{n=0}^{\infty} c_n (x - a)^n$, where $c_n$ are coefficients and $a$ is the center point.