Chapter 8: Q. 64 (page 681)

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

Short Answer

Expert verified

The series converges for every x .

Step by step solution

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

Short Answer

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 forJ0(x)?

Short Answer

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Mechanics Maths

Logic and Functions

Geometry

Study anywhere. Anytime. Across all devices.

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