Chap11_Sec6

# Chap11_Sec6 - 11 INFINITE SEQUENCES AND SERIES INFINITE...

This preview shows pages 1–14. Sign up to view the full content.

11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES

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

View Full Document
11.6 Absolute Convergence and the Ratio and Root tests In this section, we will learn about: Absolute convergence of a series and tests to determine it. INFINITE SEQUENCES AND SERIES
ABSOLUTE CONVERGENCE Given any series Σ a n , we can consider the corresponding series whose terms are the absolute values of the terms of the original series. 1 2 3 1 ... n n a a a a = = + + +

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

View Full Document
ABSOLUTE CONVERGENCE A series Σ a n is called absolutely convergent if the series of absolute values Σ | a n | is convergent. Definition 1
ABSOLUTE CONVERGENCE Notice that, if Σ a n is a series with positive terms, then | a n | = a n. So, in this case, absolute convergence is the same as convergence.

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

View Full Document
ABSOLUTE CONVERGENCE The series is absolutely convergent because is a convergent p -series ( p = 2). Example 1 1 2 2 2 2 1 ( 1) 1 1 1 1 ... 2 3 4 n n n - = - = - + - + 1 2 2 2 2 2 1 1 ( 1) 1 1 1 1 1 ... 2 3 4 n n n n n - = = - = = + + + +
ABSOLUTE CONVERGENCE We know that the alternating harmonic series is convergent. See Example 1 in Section 11.5. Example 2 1 1 ( 1) 1 1 1 1 ... 2 3 4 n n n - = - = - + - +

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

View Full Document
However, it is not absolutely convergent because the corresponding series of absolute values is: This is the harmonic series ( p -series with p = 1) and is, therefore, divergent. ABSOLUTE CONVERGENCE Example 2 1 1 1 ( 1) 1 1 1 1 1 ... 2 3 4 n n n n n - = = - = = + + + +
CONDITIONAL CONVERGENCE A series Σ a n is called conditionally convergent if it is convergent but not absolutely convergent. Definition 2

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

View Full Document
ABSOLUTE CONVERGENCE Example 2 shows that the alternating harmonic series is conditionally convergent. Thus, it is possible for a series to be convergent but not absolutely convergent. However, the next theorem shows that absolute convergence implies convergence.
ABSOLUTE CONVERGENCE If a series Σ a n is absolutely convergent, then it is convergent. Theorem 3

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

View Full Document
Observe that the inequality is true because | a n | is either a n or – a n. 0 2 n n n a a a + ABSOLUTE CONVERGENCE Theorem 3—Proof
a n is absolutely convergent, then Σ | a n | is convergent. So, Σ 2| a n | is convergent. Thus, by the Comparison Test, Σ ( a n + | a n |) is convergent. ABSOLUTE CONVERGENCE

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 01/06/2012 for the course MATH 2414.S01 taught by Professor Alans.grave during the Fall '11 term at Collins.

### Page1 / 61

Chap11_Sec6 - 11 INFINITE SEQUENCES AND SERIES INFINITE...

This preview shows document pages 1 - 14. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online