What is a Convergent Series?

A convergent series is a type of infinite series in mathematics that has a finite sum. In other words, as more terms of the series are added together, the total sum approaches a specific value. A convergent series can be thought of as a sequence of partial sums that eventually reach a fixed limit.

Criteria for Convergence

For a series to be convergent, the limit of the terms of the series as n approaches infinity must exist and be finite. Mathematically, this can be written as:

lim n→∞ an = L

Where 'an' represents the nth term of the series and 'L' is the finite limit as n approaches infinity.

Examples of Convergent Series

One of the most well-known examples of a convergent series is the geometric series. The sum of a geometric series can be calculated using the formula:

S = a / (1 - r)

Where 'a' is the first term of the series and 'r' is the common ratio between consecutive terms.

Importance of Convergent Series

Convergent series play a crucial role in calculus, analysis, and other branches of mathematics. Understanding convergent series helps mathematicians analyze functions, sequences, and other mathematical structures that involve infinite sums.