Q. 66 - 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 $J_1\left(x\right)$? 

Step-by-Step Solution

Answer

The series is converges 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 of convergence

The $J_1\left(x\right)$ is

$J_1\left(x\right)=\sum _{k=0}^{\infty }\frac{(-1{)}^k}{k!(k+1)!2^{2k+1}}{x}^{2k+1}$

We have to do ratio test first for the absolute convergence,

Assume that $b_k=\frac{(-1{)}^k}{k!(k+1)!2^{2k+1}}{x}^{2k+1}$

$$\begin{gathered} b_{k+1}=\frac{(-1{)}^{k+1}}{(k+1)!(k+1+1)!2^{2(k+1)+1}}{x}^{2(k+1)+1} \\ =\frac{(-1{)}^{k+1}}{(k+1)!(k+2)!2^{2k+3}}{x}^{2k+3} \end{gathered}$$

It implies that 

$$\begin{gathered} \underset{k\to \infty }{\lim }\left|\frac{b_{k+1}}{b_k}\right|=\underset{k\to \infty }{\lim }\left|\frac{\frac{(-1{)}^{k+1}}{(k+1)!(k+2)!2^{2k+3}}{x}^{2k+3}}{\frac{(-1{)}^k}{k!(k+1)!2^{2k+1}}{x}^{2k+1}}\right| \\ =\underset{k\to \infty }{\lim }\left|-\frac{1}{(k+1)(k+2)4}{x}^2\right| \end{gathered}$$

Calculate for k tending to infinity,

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

Limit is zero independently of x. So the series is converges for all values of x.