http://www.ck12.org Chapter 8. Infinite Series
Example 4Convergence of∑(cn)nis determined with limn→∞|cn|by the Root Test. For example∑(n
√
3 −n√
2 )n,
converges since limn→∞(√n 3 −√n 2 ) = 1 − 1 =0, but∑(√n 2 − 1 )(no exponentn) diverges by rationalizing the
numerator:
√n 2 − 1 = 2 − 1
2 n−n^1 + 2 n−n^2 +...+ 1 ≥
2 − 1
n. 2 n−n^1 =12 n√
n^2 and applying Limit Comparison test with∑ 21 n.
The Root Test is inconclusive on∑( 1 +^1 n)n, but the simpler Test for Divergence confirms its divergence since
( 1 +^1 n)n>1 always.
Example 5∑(−^1 n)qn−^1 is convergent forq>0 by the Alternating Series Test since limn→∞n^1 q=0 andn^1 q≥(n+^11 )q. It
is absolutely convergent forq>1 by thep−test. So it is conditionally convergent for 0<q≤1.
Example 6∑∞n= 1 n.n^2 !n=∑∞n= (^2) (^2 n.−^2 n 1 −)^1 != (^2) ∑∞n= (^12) nn!is absolutely convergent by the Ratio Test since limn→∞(n^2 +n+ 11 )!. 2 nn!=
limn→∞n+^21 =0.
Example 7Consider the series∑
√n+ 1
n^2 − 10 n+ 1. Noticen^2 −^10 n+1 is never 0 and is positive forn≥10, we could
ignore the terms beforen=10. Dropping the lower powers ofnleads to the candidate∑
√n
n^2 =∑n^132 for applying
Limit Comparison Test since limn→∞
√n+ 1
n^2 − 10 n+ 1 .n
√^2
n=1. So the series is (absolutely) convergent by thep−test. A
combination of tests is applied.
Multimedia Links
For video presentations of the varying tests for convergence(23.0), see Just Math Tutoring, Using the Ratio Test to
Determine if a Series Converges (7:38)
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/614; Just Math Tutoring, Root Tests for Series (10:07) 4
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/615; Just Math Tutoring, Limit Comparison Test and Direct Comparison Test (7:35)