Understanding Series in Algebra
A series is the sum of the terms of a sequence. Series are important in algebra for understanding patterns and summing values.
What is a Series?
A series is the sum of the terms in a sequence.
It can be finite or infinite.
Types of Series
- Arithmetic series: The difference between terms is constant.
- Geometric series: Each term is multiplied by a constant to get the next.
Formulas
- Arithmetic: \( S_n = \frac{n}{2}(a_1 + a_n) \)
- Geometric: \( S_n = a_1 \frac{1 - r^n}{1 - r} \) for \( r \neq 1 \)
Infinite Series
An infinite series has an infinite number of terms.
Some infinite series converge to a value; others diverge.
Practice Problems
- Find the sum of the first 10 terms of the arithmetic series: 2, 5, 8, ...
- Find the sum of the geometric series: 3, 6, 12, 24 (4 terms)
- Does the series \( 1 + \frac{1}{2} + \frac{1}{4} + \cdots \) converge?
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