Infinite series

Definition

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

5 Must Know Facts For Your Next Test

1. An infinite series converges if the sequence of partial sums approaches a finite limit.
2. The geometric series $\sum_{n=0}^{\infty} ar^n$ converges if $|r| < 1$ and its sum is $\frac{a}{1 - r}$.
3. The harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges.
4. The ratio test can determine the convergence or divergence of an infinite series by evaluating $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$.
5. The alternating series test states that an alternating series converges if its terms decrease in absolute value and approach zero.

Review Questions

• What condition must be met for a geometric series to converge?
• Explain why the harmonic series diverges.
• How does the ratio test help in determining the convergence of an infinite series?

Related terms

Geometric Series: A series of the form $\sum_{n=0}^{\infty} ar^n$, where each term is obtained by multiplying the previous term by a fixed non-zero number $r$.

Harmonic Series: $\sum_{n=1}^{\infty} \frac{1}{n}$, which is known to diverge because its partial sums grow without bound.

Ratio Test: A test for determining the convergence or divergence of a series by examining the limit $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$.