The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.

Home Infinite Series Absolute and Conditional Convergence Theorem 1: Convergence of Absolutly Convergent Series
 
del.icio.us Reddit Facebook StumbleUpon MySpace Twitter Search the VIAS Library  |  Index

Theorem 1: Convergence of Absolutly Convergent Series

THEOREM 1

Every absolutely convergent series is convergent. That is, if the absolute value series 09_infinite_series-322.gif converges, then 09_infinite_series-323.gif converges.

Discussion

This theorem shows that if a positive term series 09_infinite_series-324.gif 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 09_infinite_series-325.gif, 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 09_infinite_series-326.gif converges and

let

bn = an + |an|.

Then

an = bn - |an|

and

09_infinite_series-327.gif

(See Figure 9.6.1).

09_infinite_series-333.gif

Figure 9.6.1

Both 09_infinite_series-328.gif and 09_infinite_series-329.gif have nonnegative terms.

Moreover, 09_infinite_series-330.gif converges and bn ≤ 2 |an|. By the Comparison Test,

09_infinite_series-331.gif

converges. Then using the Sum and Constant Rules,

09_infinite_series-332.gif

converges.

Example 1: Absolute Convergence
Example 2: Conditional Convergence

Home Infinite Series Absolute and Conditional Convergence Theorem 1: Convergence of Absolutly Convergent Series

Last Update: 2006-11-07