Infinite Series
Functions & Advanced AlgebraAn infinite series is the sum of infinitely many terms of a sequence, which may converge to a finite value or diverge.
Definition
An infinite series is what you get when you add up infinitely many numbers following a pattern. Sometimes the total approaches a specific finite number; sometimes it grows forever.
Example
$1 + 1/2 + 1/4 + 1/8 + \ldots$ keeps going forever but sums to $2$. In contrast, $1 + 1 + 1 + 1 + \ldots$ also goes forever and the total grows without end.
Key Insight
It seems impossible that adding infinitely many things could give a finite answer, but if the terms get small enough fast enough, it can happen. The key is whether the running total settles down.
Definition
An infinite series $\sum_{n=1}^{\infty} a_n$ is defined as the limit of its partial sums: if $\lim_{n \to \infty} S_n = L$, the series converges to $L$; otherwise it diverges. The terms $a_n$ must approach $0$ for convergence, but this is not sufficient.
Example
$\sum 1/n^2 = 1 + 1/4 + 1/9 + 1/16 + \ldots$ converges to $\pi^2/6 \approx 1.645$. $\sum 1/n = 1 + 1/2 + 1/3 + \ldots$ diverges (harmonic series). Both have terms approaching $0$, but behavior differs.
Key Insight
Infinite series are fundamental to calculus (Taylor/Maclaurin series represent functions as polynomials), Fourier analysis, and physics. They let us compute exact values for irrational quantities like $\pi$ and $e$.
Definition
An infinite series is a formal expression $\sum a_n$; its value (if it exists) is $\lim S_n$ in the underlying metric space. In a Banach space, absolute convergence ($\sum \|a_n\| < \infty$) implies convergence. Power series $\sum a_n x^n$ converge on open disks in $\mathbb{C}$ and represent analytic functions within their radius of convergence.
Example
The exponential function: $e^x = \sum_{n=0}^{\infty} x^n/n!$ converges for all $x \in \mathbb{C}$. This power series representation enables analytic continuation, defining $e^z$ for complex $z$ and connecting to Euler's identity.
Key Insight
The study of infinite series gave rise to modern real analysis: Cauchy, Weierstrass, and Riemann formalized convergence to resolve paradoxes and provide rigorous foundations for calculus.