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Various tests to determine the convergence or divergence of power series in BC Calculus, including the Nth-Term Test, Direct Comparison Test, and Ratio Test. It also discusses the definitions of absolute and conditional convergence and the Convergence Theorem for Power Series. The document concludes by finding the radius and interval of convergence for several power series.
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10.4 Radius of Convergence
Convergence for a :
Geometric Series: Maclaurin series for sin x , cos x , ex :
Recall from 10.1 infinite series:
The nth-term Test for Divergence If (^) n limโโ an =/ 0 , then โ diverges. *The converse of this is not true. If the
โ n =
an (^) n limโโ an = 0 series could converge.
โ n =
2 n^ b. โ
โ n =
n! 2 n !+1 c.^ โ
โ n =1 n
1
10.4 Radius of Convergence
The direct comparison test is a tool that we can use to determine convergence for complicated, positive series by comparing them with simpler series.
Direct Comparison Test Let 0 < an < bn for all n
โ n =
bn โ
โ n =
an
โ n =
an โ
โ n =
bn
โ n =
1 2+3 n^ b.^ โ
โ n =
x^2 n ( n !)^2 c.^ โ
โ n =
x^2 n n !+
10.4 Radius of Convergence
Ratio test Let โ an be a series with nonzero terms.
The ratio test is particularly useful for series that converge rapidly (i.e. factorials or exponentials).
โ n =0 n!
2 n b. โ^ โ n =0 3
n^2 2 n n +
10.4 Radius of Convergence
c. โ
โ n =1 n!
nn
d. โ(โ )
โ n =
1 n^ n โ+1 n
10.4 Radius of Convergence
c. โ
โ n =0 n^^2
(โ1) ( nx โ2) n +1 n d. โ
โ n =
( x โ3) n + ( n +1) 4 n +
