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## Theorem 1: Convergence of Absolutly Convergent Series

THEOREM 1

Every absolutely convergent series is convergent. That is, if the absolute value series converges, then converges.

Discussion

This theorem shows that if a positive term series is convergent,

then it remains convergent if we make some or all of the terms bn negative, because the new series will still be absolutely convergent.

Given an arbitrary series , the theorem shows that exactly one of the

following three things can happen:

The series is absolutely convergent. The series is conditionally convergent. The series is divergent.

PROOF OF THEOREM 1

We use the Sum Rule. Assume converges and

let

bn = an + |an|.

Then

an = bn - |an|

and

(See Figure 9.6.1).

Figure 9.6.1

Both and have nonnegative terms.

Moreover, converges and bn ≤ 2 |an|. By the Comparison Test,

converges. Then using the Sum and Constant Rules,

converges.

 Example 1: Absolute Convergence
 Example 2: Conditional Convergence

Last Update: 2006-11-07