Summation Convergence Calculator

The Summation Convergence Calculator is designed to determine whether a given series converges or diverges. Currently, it supports geometric series, with the potential to expand to arithmetic and harmonic series.

Overview of Series Convergence

In mathematics, the convergence of a series refers to whether the sum of the infinite sequence approaches a finite value. Different types of series have different criteria for convergence.

Geometric Series Convergence

A geometric series converges when the absolute value of the common ratio is less than 1. The sum of the infinite series is given by:

\[ \text{Sum} = \frac{a}{1 - r} \]

where \(a\) is the first term and \(r\) is the common ratio.

Example Calculation

If the first term \(a = 5\) and the common ratio \(r = 0.5\), the series converges to:

\[ \text{Sum} = \frac{5}{1 - 0.5} = 10 \]

If \(r\) is equal to or greater than 1 (in absolute value), the series diverges.

Importance

Understanding the convergence of series is crucial in fields such as calculus, engineering, and physics, where series are used to model various phenomena.

Common FAQs

1. What is a series?

   • A series is the sum of the terms of a sequence.

2. What does it mean for a series to converge?

   • A series converges when its sum approaches a finite value as more terms are added.

3. Can all series converge?

   • No, some series diverge, meaning their sum does not approach a finite value.