Infinite Series

An infinite series is a summation across an infinite sequence $(𝑎 𝑛) \infty 𝑛 = 1$ , where if $𝑠 𝑛$ is the partial sum up to the $𝑛$ th term

𝑠 = \infty \sum 𝑘 = 1 𝑎 𝑘 = l i m 𝑛 \to \infty 𝑠 𝑛 = l i m 𝑛 \to \infty 𝑛 \sum 𝑘 = 1 𝑎 𝑘

An infinite series is said to be convergent precisely when the above limit converges, and divergent when the limit does not exist. See Tests for series divergence. Another important concept is Absolute and conditional convergence.

Examples

Well known infinite series include

The Geometric series $\sum \infty 𝑛 = 0 𝑟 𝑛$ , convergent for $| 𝑟 | < 1$ with value $𝑠 = 1 1 - 𝑟$
The 𝑝-series $\sum \infty 𝑛 = 1 1 𝑛 𝑝$ , convergent for $𝑝 > 1$ .
Alternating series are a group of series which alternate between positive and negative terms.
Power series are a ubiquitous way of constructing $ℵ 0$ -order polynomials using an underlying infinite sequence.
Fourier series are a way to represent periodic functions as the sum of sines and cosines.