What is "absolute convergence"

In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute value of the summand is finite . More precisely, a real or complex series $\textstyle\sum_{n=0}^\infty a_n$ is said to converge absolutely if $\textstyle\sum_{n=0}^\infty \left|a_n\right| = L$ for some real number $\textstyle L$. Similarly, an improper integral of a function, $\textstyle\int_0^\infty f(x)\,dx$, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if $\textstyle\int_0^\infty \left|f(x)\right|dx = L.$

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, yet is broad enough to occur commonly. (A convergent series that is not absolutely convergent is called conditionally convergent.)