# PP 12.6 - Absolute Convergence and the Ratio and Root Tests...

This preview shows pages 1–7. Sign up to view the full content.

Absolute Convergence and the Ratio and Root Tests

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Definition A series Σ a n is called absolutely convergent if the series of absolute values Σ | a n | is convergent. n n = - 1 2 1 : Example = = = - 1 1 2 1 2 1 n n n n Both series are convergent geometric series by the Geometric Series Test. Hence, the original series is absolutely convergent.
Example = - 1 ) 1 ( n n n We showed in a previous section that the alternating harmonic series is convergent. = = = - 1 1 1 ) 1 ( n n n n n We used the p-series test to show that the harmonic series is divergent. Hence, not all convergent series are absolutely convergent.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Definition A series Σ a n is called conditionally convergent if it is convergent but not absolutely convergent. We just showed that the alternating harmonic series is conditionally convergent.
Absolute Convergence Test If a series Σ a n is absolutely convergent, then it is convergent. Proof: n n n a a a 2 0 + convergent convergent absolutely n n a a convergent 2 n a ( 29 convergent + n n a a by the Comparison Test.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
We know that a sum of convergent series is convergent, so ( 29 . convergent is - + = n n n n a a a a Hence, series are conditionally convergent,
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 01/21/2011 for the course PHYS 4A 60865 taught by Professor L. oldewurtel during the Fall '09 term at Irvine Valley College.

### Page1 / 23

PP 12.6 - Absolute Convergence and the Ratio and Root Tests...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online