# Real series/Absolutely convergent/Section

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

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 .