Absolute convergence

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In mathematics, a series or integral is said to converge absolutely if the sum or integral of the absolute value of the summand or integrand is finite.

More precisely, a series \sum_{n=0}^\infty a_n is said to converge absolutely if and only if \sum_{n=0}^\infty \left|a_n\right| < \infty.

Likewise, an integral \int_A f(x)\,dx is said to converge absolutely if and only if \int_A \left|f(x)\right|\,dx < \infty.

The property of absolute convergence is important because it is generally required in order for rearrangements and products of sums to work in an intuitive fashion.

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The theorem regarding the sum \sum_{n=0}^\infty a_n can be quickly proven, in the case that for every n, the matching term an in the sum is a complex number: The sum of all the terms an can be interpreted as a vector addition path through the complex plane. If the length of the path, which is the sum of all the lengths of the parts | an | , is finite, then the end point has to be a finite distance from the origin.

Absolute convergence means that the value of the sum/integral is independent of the order in which the sum is performed. That is, any rearrangement of the series

\sum_{n=0}^\infty a_{\sigma(n)}

(where σ is some permutation of the natural numbers) will not alter the value to which the series converges, although it may change how quickly the final sum is approached.

A similar result applies to integrals. Indeed: in Lebesgue's theory of integration, sums may be treated as special cases of integrals, rather than as a separate case.

The Cauchy product of two series converges to the product of the sums if at least one of the series converges absolutely. That is, suppose:

\sum_{n=0}^\infty a_n = A
\sum_{n=0}^\infty b_n = B.

The Cauchy product is defined as the sum of terms cn where:

c_n = \sum_{k=0}^n a_k b_{n-k}.

Then, if either the an or bn sum converges absolutely, then

\sum_{n=0}^\infty c_n = AB.

  • Walter Rudin, Principles of Mathematical Analysis (McGraw-Hill: New York, 1964).
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