# Interactive Real Analysis

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## 4.1. Series and Convergence

### Theorem: Absolute Convergence implies Convergence

If a series converges absolutely, it converges in the ordinary sense. The converse is not true.

Suppose is absolutely convergent. Let

Tj = |a1| + |a2| + ... + |aj|

be the sequence of partial sums of absolute values, and

Sj = a1 + a2 + ... + aj

be the "regular" sequence of partial sums. Since the series converges absolutely, there exists an integer N such that:

| Tn - Tm| = |an| + |an-1| + ... + |am+2| + |am+1| <

if n > m > N. But we have by the triangle inequality that

| Sn - Sm| = | an + an-1 + ... + am+2 + am+1 |
|an| + |an-1| + ... + |am+2| + |am+1| = | Tn - Tm | <

Hence the sequence of regular partial sums {Sn} is Cauchy and therefore must converge (compare this proof with the Cauchy Criterion for Series).

The converse is not true because the series converges, but the corresponding series of absolute values does not converge.

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