Series

Definition

A series is the sum of the terms of a sequence. It can be finite or infinite, and is often expressed using summation notation.

5 Must Know Facts For Your Next Test

1. The notation for a series is usually given by $\sum_{i=1}^{n} a_i$, where $a_i$ are the terms of the sequence.
2. An arithmetic series has a constant difference between consecutive terms, while a geometric series has a constant ratio.
3. The formula for the sum of an arithmetic series is $S_n = \frac{n}{2}(a_1 + a_n)$, where $S_n$ is the sum, $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the last term.
4. The formula for the sum of a geometric series is $S_n = a \frac{1-r^n}{1-r}$ for $|r| < 1$, where $S_n$ is the sum, $a$ is the first term, and $r$ is the common ratio.
5. Convergence in infinite series: An infinite geometric series converges if its common ratio's absolute value is less than one ($|r| < 1$).

Review Questions

• What is the general form for expressing a series in summation notation?
• How do you find the sum of an arithmetic series with given first and last terms?
• Under what condition does an infinite geometric series converge?

Related terms

Sequence:

An ordered list of numbers that often follows some specific pattern or rule.

Arithmetic Series:

A series where each term after the first differs from its predecessor by a constant amount.

Geometric Series:

A series with a constant ratio between successive terms.