Explanation Given: F ( x ) = ∑ k = 0 ∞ x 2 k 2 k . k ! Proof: (a) Consider a n = x 2 n 2 n n ! and a n + 1 = x 2 ( n + 1 ) 2 n + 1 ( n + 1 ) ! So, the ratio r = lim n → ∞ | a n + 1 a n | = lim n → ∞ | x 2 ( n + 1 ) 2 n + 1 ( n + 1 ) ! . 2 n n ! x 2 n | = lim n → ∞ | x 2 n . x 2 2.2 n ( n + 1 ) n ! . 2 n n ! x 2 n | = lim n → ∞ | x 2 2 ( n + 1 ) | = | x 2 | lim n → ∞ ( 1 2 ( n + 1 ) ) = 1 2 x 2 .0 = 0 Since, r = 0 for ∀ x . Therefore, the radius of convergence ( R = 1 r ) of F ( x ) = ∑ k = 0 ∞ x 2 k 2 k . k ! is ∞ Thus, it is proved that the radius of convergence F ( x ) = ∑ k = 0 ∞ x 2 k 2 k . k ! is ∞ (b) Consider y ' ' = x y ' + y ... ( 1 ) y ( 0 ) = 1 , y ' ( 0 ) = 0 Let y = ∑ n = 0 ∞ a n x n be a solution of (1) So, y ' = ∑ n = 0 ∞ a n n x n − 1 y ' ' = ∑ n = 0 ∞ a n n ( n − 1 ) x n − 2 Putting these values in equation (1), we get ∑ n = 0 ∞ a n n ( n − 1 ) x n − 2 = ∑ n = 0 ∞ x ( a n n x n − 1 ) + ∑ n = 0 ∞ a n x n ...

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 11, Problem 109CRE

To determine

To prove:

(a) $F (x)$ has infinite radius of convergence.

(b) $y = F (x)$ is a solution of $y^{''} = x y^{'} + y, y (0) = 1, y^{'} (0) = 0$

(c) Plot the partial sums $S_{N}$ for $N = 1, 3, 5, 7...$ on the same set of axes.

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Chapter 11, Problem 108CRE

Chapter 11, Problem 110CRE

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Chapter 11 Solutions

CALCULUS (CLOTH)

Ch. 11.1 - Prob. 1PQ Ch. 11.1 - Prob. 2PQ Ch. 11.1 - Prob. 3PQ Ch. 11.1 - Prob. 4PQ Ch. 11.1 - Prob. 5PQ Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E

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To prove: (a) F ( x ) has infinite radius of convergence. (b) y = F ( x ) is a solution of y ' ' = x y ' + y , y ( 0 ) = 1 , y ' ( 0 ) = 0 (c) Plot the partial sums S N for N = 1 , 3 , 5 , 7... on the same set of axes.

Solution Summary: The author proves that F (x) has infinite radius of convergence.

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To prove:

(a) $F (x)$ has infinite radius of convergence.

(b) $y = F (x)$ is a solution of $y^{''} = x y^{'} + y, y (0) = 1, y^{'} (0) = 0$

(c) Plot the partial sums $S_{N}$ for $N = 1, 3, 5, 7...$ on the same set of axes.

Want to see the full answer?

Chapter 11 Solutions

To prove: (a) F ( x ) has infinite radius of convergence. (b) y = F ( x ) is a solution of y ' ' = x y ' + y , y ( 0 ) = 1 , y ' ( 0 ) = 0 (c) Plot the partial sums S N for N = 1 , 3 , 5 , 7... on the same set of axes.

Solution Summary: The author proves that F (x) has infinite radius of convergence.

Concept explainers

Videos

To prove: (a) F(x) has infinite radius of convergence. (b) y=F(x) is a solution of y''=xy'+y,y(0)=1,y'(0)=0 (c) Plot the partial sums SN for N=1,3,5,7... on the same set of axes.

Want to see the full answer?

Chapter 11 Solutions

To prove:

(a) $F (x)$ has infinite radius of convergence.

(b) $y = F (x)$ is a solution of $y^{''} = x y^{'} + y, y (0) = 1, y^{'} (0) = 0$

(c) Plot the partial sums $S_{N}$ for $N = 1, 3, 5, 7...$ on the same set of axes.