Alternating Series Test

# Alternating Series Test - A Caution on the Alternating...

This preview shows pages 1–2. Sign up to view the full content.

A Caution on the Alternating Series Test Theorem 14 (The Alternating Series Test) of the textbook says: The series X n =1 ( - 1) n +1 u n = u 1 - u 2 + u 3 - u 4 + ··· converges if all of the following conditions are satisﬁed: 1. u n > 0 for all n N . 2. u n u n +1 for all n N , for some integer N . 3. u n 0 as n → ∞ . Clearly, to show the convergence, you need to check all these three conditions. What can you conclude if a given alternating series fails one or more of the three conditions? Is that series divergent? Unfortunately, the failure of this test does not immediately lead to the divergence! You need to use some other test, often, the n th Term Test for Divergence to conclude that the series diverges. The reason is the following. The equivalent statement to Theorem 14 is that: (1) “If a given alternating series diverges, then it fails to satisfy one or more of the above three conditions.” This is diﬀerent from the following statement, which is false : (2)

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 04/21/2008 for the course MATH 021C taught by Professor Saito during the Spring '08 term at UC Davis.

### Page1 / 2

Alternating Series Test - A Caution on the Alternating...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online