   Chapter 10.4, Problem 52E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem

# Finding the Radius of Convergence In Exercises 49–52, find the radius of convergence of (a) f(x), (b) f ' ( x ) , (c) f " ( x ) , and (d) ∫ f ( x )   d x . f ( x ) = ∑ n = 1 ∞ ( − 1 ) n + 1 ( x − 1 ) n n

(a)

To determine

To calculate: The radius of convergence of the power series f(x)=n=1(1)n+1(x1)nn.

Explanation

Given Information:

The provided function is f(x)=n=1(1)n+1(x1)nn.

Formula used:

Ratio test:

For an infinite series n=1an with nonzero terms.

(a) If limn|an+1an|<1 then the series converges.

(b) If limn|an+1an|>1 or limn|an+1an|= then the series diverges.

(c) If limn|an+1an|=1 then test is inconclusive.

Convergence of a Power Series:

If a power series is centered at a constant c, then either of the following condition is true.

(a) The radius of convergence R=0 if the series converges only at c.

(b) The radius of convergence R exist and is a positive real number such that series converges for |xc|<R and diverges for |xc|>R.

(c) The radius of convergence R= if the series converges for all x.

Calculation:

Consider the function f(x)=n=1(1)n+1(x1)nn

Apply the ratio test as,

limn|an

(b)

To determine

To calculate: The radius of convergence of f(x) where f(x)=n=1(1)n+1(x1)nn.

(c)

To determine

To calculate: The radius of convergence of f(x) where f(x)=n=1(1)n+1(x1)nn.

(d)

To determine

To calculate: The radius of convergence of f(x)dx where the function f(x)=n=1(1)n+1(x1)nn.

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