Calc 2 Lecture NotesSection 8.5Page 1 of 3
Section 8.5: Absolute Convergence and the Ratio Test
Big idea: If a series has some negative terms (but the terms do not necessarily alternate in sign), and converges, then the series will converge because those occasional negative terms will “cancel” some of the positive terms, thus hastening the convergence overall.
Big skill: You should be able to show when a series converges absolutely, and also be able to apply the root and ratio tests.
An absolutely convergent series has the property that not only does converge (the idea being that the series contains negative terms which help it converge) , but converges also.
A conditionally convergent series has the property that converges, but diverges.
Theorem 5.1: Absolute Convergence Implies Overall Convergence
If converges, then converges.
Practice:
Determine the convergence of .
Determine the convergence of .
Theorem 5.2: The Ratio Test
Given , with ak 0 for all k, suppose that . Then:
(i).if L < 1, the series converges absolutely.
(ii).if L > 1 (or L = ), the series diverges.
(iii).if L =1, no conclusion can be made.
Practice:
Determine the convergence of .
Determine the convergence of .
Theorem 5.3: The Root Test
Given , with ak 0 for all k, suppose that . Then:
(i).if L < 1, the series converges absolutely.
(ii).if L > 1 (or L = ), the series diverges.
(iii).if L =1, no conclusion can be made.
Practice:
Determine the convergence of .
Determine the convergence of .
Test / When to Use / Conclusions / SectionGeometric Series / / Converges to if and diverges if . / 8.2
Kth-Term Test / All series. / If , the series diverges. / 8.2
Integral Test / , where , f is continuous and decreasing, and
f(x) 0 / and both converge or both diverge. / 8.3
p-Series / / Converges if p > 1 and diverges if p 1. / 8.3
Comparison Test / 0 akbk for all k . / If converges, then converges.
If diverges, then diverges. / 8.3
Limit Comparison Test / ak, bk > 0, and
/ and both converge or both diverge. / 8.3
Alternating Series Test / , where / If and , then the series converges. / 8.4
Absolute Convergence / Series with some positive and some negative terms (including alternating series) / If converges, then converges absolutely. / 8.5
Ratio Test / Any series (especially those with exponentials or factorials) / If and if L < 1, the series converges absolutely. / 8.5
Root Test / Any series (especially those with exponentials) / If and if L < 1, the series converges absolutely. / 8.5