Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a series of numbers is said to converge absolutely if the sum (or integral) of the absolute value of the summand or integrand is finite. More precisely, a real or complex series sum_ a_n is said to converge absolutely if sum_ Absolute convergence is important to the study of infinite series because its definition is strong enough to have properties of...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a series of numbers is said to converge absolutely if the sum (or integral) of the absolute value of the summand or integrand is finite. More precisely, a real or complex series sum_ a_n is said to converge absolutely if sum_ Absolute convergence is important to 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. If the metric d on G is complete, then every absolutely convergent series is convergent. The proof is the same as for complex-valued series: use the completeness to derive the Cauchy criterion for convergence—a series is convergent if and only if its tails can be made arbitrarily small in norm—and apply the triangle inequality. In particular, for series with values in any Banach space, absolute convergence implies convergence. The converse is also true: if absolute convergence implies convergence in a normed space, then the space is a Banach space.
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