Real series/Absolutely convergent/Section

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A series

of real numbers is called absolutely convergent, if the series


Let be given. We use the Cauchy-criterion. Since the series converges absolutely, there exists some such that for all the estimate

holds. Therefore,

which means the convergence.

A convergent series does not in general converge absolutely, the converse of fact does not hold. Due to the Leibniz criterion, the alternating harmonic series

converges, and its sum is , a result we can not prove here. However, the corresponding absolute series is just the harmonic series, which diverges due to example.

The following statement is called the direct comparison test.

Let be a convergent series of real numbers and a sequence of real numbers fulfilling for all . Then the series

is absolutely convergent.

This follows directly from the Cauchy-criterion.

We want to determine whether the series

converges. We use the direct comparison test and example, where we have shown the convergence of . For we have

Hence, converges and therefore also . This does not say much about the exact value of the sum. With much more advanced methods, one can show that this sum equals .