## 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

T_{j}= |a_{1}| + |a_{2}| + ... + |a_{j}|

be the sequence of partial sums of absolute values, and

S_{j}= a_{1}+ a_{2}+ ... + a_{j}

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

| T_{n}- T_{m}| = |a_{n}| + |a_{n-1}| + ... + |a_{m+2}| + |a_{m+1}| <

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

| S_{n}- S_{m}| = | a_{n}+ a_{n-1}+ ... + a_{m+2}+ a_{m+1}|

|a_{n}| + |a_{n-1}| + ... + |a_{m+2}| + |a_{m+1}| = | T_{n}- T_{m}| <

Hence the sequence of regular partial sums *{S _{n}}* 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.