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.

5 Must Know Facts For Your Next Test

1. The general form of an infinite series is $\sum_{n=1}^{\infty} a_n$ where $a_n$ are the terms of the sequence.
2. A series converges if the sum approaches a specific value as more terms are added, otherwise it diverges.
3. The geometric series $\sum_{n=0}^{\infty} ar^n$ converges if $|r| < 1$ and its sum is $\frac{a}{1-r}$.
4. The harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges even though its terms approach zero.
5. Convergence tests such as the Ratio Test and Root Test are used to determine whether an infinite series converges or diverges.

Review Questions

• What is the condition for convergence of a geometric series?
• Explain why the harmonic series diverges.
• Describe one method used to test for the convergence of an infinite series.

Related terms

Convergence: A property of an infinite series where its partial sums approach a specific value as more terms are added.

Divergence: A property of an infinite series where its partial sums do not approach any specific limit as more terms are added.

$\Sigma$ Notation: $\Sigma$ (sigma) notation is a way to write sums in a concise form using upper and lower bounds to define the range of summation.