A series in mathematics is the sum of the terms of a sequence. It can be either finite, with a limited number of terms, or infinite, where terms continue indefinitely. Series are often represented using sigma notation and can be convergent (approaching a specific value) or divergent (growing without bound).
To determine if a series is convergent or divergent, you can use various tests such as the comparison test, ratio test, root test, integral test, and alternating series test. Each test has specific criteria that help assess the behavior of the series as the number of terms increases.
A sequence is an ordered list of numbers, where each number is called a term. A series, on the other hand, is the sum of the terms of a sequence. While a sequence lists the terms individually, a series adds them together to form a single value or expression.
Common types of series include arithmetic series, geometric series, harmonic series, and power series. Each type has specific characteristics and formulas for finding the sum of its terms. For example, a geometric series has a constant ratio between consecutive terms, while an arithmetic series has a constant difference.
No, a divergent series does not have a finite sum. By definition, a divergent series grows without bound or oscillates without settling to a specific value. Only convergent series have finite sums, where the partial sums approach a specific limit as the number of terms increases.