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Fundamentals Of Differential Equations And Boundary Value Problems
Found in: Page 421
Fundamentals Of Differential Equations And Boundary Value Problems

Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

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Short Answer

In problems 1-6, determine the convergence set of the given power series.

n=13n3(x-2)n

The set is, x[1,3]

See the step by step solution

Step by Step Solution

Step 1:To Find the Radius of convergence

Use the ratio test to determine the radius of convergence.

limnanan+1=limn3n33(n+1)3=limnn3n3+3n2+3n+1×33=limnn3n3+3n2+3n+1=limn11+(3/n)+(3/n2)+(1/n3)=1

The radius of convergence is 1, therefore convergent set for the given power series is.|x2|<1

Step 2: Find the set of convergence

To completely identify the convergence set, we have to check whether the boundary points 1 and 3 are included in the set or not.

Checking at ,x=1 by substituting the value of x by 1,

n=03n3 x-2n=n=03n3 1-2n=n=03n3 -1n

The above series is an alternating power series withp=3>1 , which is convergent, thus the point 1 is included in the convergent set.

Similarly, checking at x=3, by substituting the value of x by 3,

n=03n3 x-2n=n=03n3 3-2n=n=03n3

The above series is a power series with p=3>1, which is convergent, thus the point 3 is included in the convergent set.

The convergent set for the given power series is.x[1,3]

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