Infinite Series

What is an Infinite Series?

An infinite series is the sum of an endless list of numbers, often written with dots at the end to show it keeps going. For example, $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots$ adds numbers forever.

General Form:

An infinite series is written as: $S = a_1 + a_2 + a_3 + a_4 + \dots$

where $a_1, a_2, a_3, \dots$ are the terms of the series.

Types of Infinite Series:

Convergent Series:
- The sum approaches a specific finite value as more terms are added.
- Example: The geometric series $S = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$ converges to 2.
Divergent Series:
- The sum grows indefinitely as more terms are added.
- Example: The harmonic series $S = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots$ diverges because the sum increases without bound.
Alternating Series:
- The terms alternate between positive and negative.
- Example: $S = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$ .

Applications:

Mathematics and Calculus:
- Infinite series are foundational for understanding calculus concepts like limits and integrals.
- They are used to approximate functions through Taylor and Maclaurin series.
Physics and Engineering:
- Infinite series are used to model waveforms, analyze electrical circuits, and solve differential equations.
Computer Science:
- Algorithms for numerical approximations often rely on summing infinite series for better accuracy.