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Q. 64

Found in: Page 681


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 J0(x)?

The series converges for every x .

See the step by step solution

Step by Step Solution

Step 1.Given information 

We have to find out the interval for convergence for Jp(x)

Step 2. Representation of function 

We denote the the given function as


Step 3. Finding the interval of convergence for given function  


For finding the convergence of the function we will do the ratio test for absolute convergence we will assume bk=(1)k(k!)222kx2k therefore the next term will be bk+1=(1)k+1[(k+1)!]222(k+1)x2(k+1) implies that

role="math" limkbk+1bk=limk[(k+1)!]222(k+1)x2(k+1)(1)k(k!)222kx2k=limk1(k+1)24x2

Now we will be evaluating the limit k

so, limkx21(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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