Q. 68 - Calculus

Question

The second-order differential equation  

$x^2{y}^{\text{'}\text{'}}+x{y}^{\text{'}}+\left(x^2-p^2\right)=0$

where p is a non-negative integer, arises in many applications in physics and engineering, including one model for the vibration of a beaten drum. The solution of the differential equation is called the Bessel function of order p, denoted by $J_p\left(x\right)$.It may be shown that $J_p\left(x\right)$ is given by the following power series in x:  

$J_p\left(x\right)=\sum _{k=0}^{\infty }\frac{(-1{)}^k}{k!(k+p)!2^{2k+p}}{x}^{2k+p}$

What is the interval of convergence for $J_p\left(x\right)$ where p is a non-negative integer 

Step-by-Step Solution

Answer

So series converges absolutely for all values of x.

Step 1. Given information

An expression is given as $J_p\left(x\right)=\sum _{k=0}^{\infty }\frac{(-1{)}^k}{k!(k+p)!2^{2k+p}}{x}^{2k+p}$

Step 2. Interval for non negative terms

We have to do first ratio test. Let assume $b_k=\frac{(-1{)}^k}{k!(k+p)!2^{2k+p}}{x}^{2k+p}$

Therefore,

$$\begin{gathered} b_{k+1}=\frac{(-1{)}^{k+1}}{(k+1)!(k+1+p)!2^{2(k+1)+p}}{x}^{2(k+1)+p} \\ =\frac{(-1{)}^{k+1}}{(k+1)!(k+p+1)!2^{2k+2+p}}{x}^{2k+p+2} \end{gathered}$$

Implies that

$$\begin{gathered} \underset{k\to \infty }{\lim }\left|\frac{b_{k+1}}{b_k}\right|=\underset{k\to \infty }{\lim }\left|\frac{\frac{(-1{)}^{k+1}}{(k+1)!(k+p+1)!2^{2k+p+2}}{x}^{2k+p+2}}{\frac{(-1{)}^k}{k!(k+p)!2^{2k+p}}{x}^{2k+p}}\right| \\ =\underset{k\to \infty }{\lim }\left|-\frac{1}{(k+1)(k+p+1)4}{x}^2\right| \end{gathered}$$

Now evaluate the limit when k tends to infinity as,

$\underset{k\to \infty }{\lim }\left|x^2\right|\frac{-1}{(k+1)(k+p+1)4}=0$

The value of limit is zero independently of value of x.

So series converges absolutely for all values of x.