Loterre: Mathematics: absolute convergence

mathematical analysis > calculus > integral calculus > absolute convergence

mathematical analysis > calculus > series > absolute convergence

Preferred term

absolute convergence

Definition(s)

In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series ${\\displaystyle \ extstyle \\sum _{n=0}^{\\infty }a_{n}}$ is said to converge absolutely if ${\\displaystyle \ extstyle \\sum _{n=0}^{\\infty }\\left|a_{n}\ ight|=L}$ for some real number ${\\displaystyle \ extstyle L.}$ Similarly, an improper integral of a function, ${\\displaystyle \ extstyle \\int _{0}^{\\infty }f(x)\\,dx,}$ is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if ${\\displaystyle \ extstyle \\int _{0}^{\\infty }|f(x)|dx=L.}$
Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess – a convergent series that is not absolutely convergent is called conditionally convergent, while absolutely convergent series behave "nicely". For instance, rearrangements do not change the value of the sum. This is not true for conditionally convergent series: The alternating harmonic series ${\ extstyle 1-{\ rac {1}{2}}+{\ rac {1}{3}}-{\ rac {1}{4}}+{\ rac {1}{5}}-{\ rac {1}{6}}+\\cdots }$ converges to ${\\displaystyle \\ln 2,}$ while its rearrangement ${\ extstyle 1+{\ rac {1}{3}}-{\ rac {1}{2}}+{\ rac {1}{5}}+{\ rac {1}{7}}-{\ rac {1}{4}}+\\cdots }$ (in which the repeating pattern of signs is two positive terms followed by one negative term) converges to ${\ extstyle {\ rac {3}{2}}\\ln 2.}$