Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Exercise 64-68 concern with the bessel function.
What is the interval for convergence for $J_{0} (x) ?$

The series converges for every x .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1.Given information

We have to find out the interval for convergence for $J_{p} (x)$

Step 2. Representation of function

We denote the the given function as

J_{p} (x) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{k! (k + p)! 2^{2 k + p}} x^{2 k + p}

Step 3. Finding the interval of convergence for given function

$\begin{aligned} J_{0} (x) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{k! (k + 0)! 2^{2 k + 0}} x^{2 k + 0} \\ = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{(k!)^{2} 2^{2 k}} x^{2 k} \end{aligned}$

For finding the convergence of the function we will do the ratio test for absolute convergence we will assume $b_{k} = \frac{(- 1)^{k}}{(k!)^{2} 2^{2 k}} x^{2 k}$ therefore the next term will be $b_{k + 1} = \frac{(- 1)^{k + 1}}{[(k + 1)!]^{2} 2^{2 (k + 1)}} x^{2 (k + 1)}$ implies that

role="math" $\begin{aligned} lim_{k \to \infty} |\frac{b_{k + 1}}{b_{k}}| = lim_{k \to \infty} |\frac{[(k + 1)!]^{2} 2^{2 (k + 1)}}{} x^{2 (k + 1)}| \frac{(- 1)^{k}}{(k!)^{2} 2^{2 k} x^{2 k}} ∣ \\ = lim_{k \to \infty} |- \frac{1}{(k + 1)^{2} 4} x^{2}| \end{aligned}$

Now we will be evaluating the limit $k \to \infty$

so, $lim_{k \to \infty} |x^{2}| \frac{- 1}{(k + 1)^{2}} = 0$ zero is the value no matter what is the value of x .

Step 4. The convergence interval is

Hence by the ratio test it is clear that this series absolutely converges for every value of x

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Calculus

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Short Answer

Exercise 64-68 concern with the bessel function.
What is the interval for convergence for $J_{0} (x) ?$

Step by Step Solution

Step 1.Given information

Step 2. Representation of function

Step 3. Finding the interval of convergence for given function

Step 4. The convergence interval is

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Short Answer

Exercise 64-68 concern with the bessel function.What is the interval for convergence for J0(x)?

Step by Step Solution

Step 1.Given information

Step 2. Representation of function

Step 3. Finding the interval of convergence for given function

Step 4. The convergence interval is

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Exercise 64-68 concern with the bessel function.
What is the interval for convergence for $J_{0} (x) ?$