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Q. 66

Expert-verifiedFound in: Page 681

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

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 ${J}_{1}\left(x\right)$?

The series is converges for all values of *x.*

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

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 ${b}_{k}=\frac{(-1{)}^{k}}{k!(k+1)!{2}^{2k+1}}{x}^{2k+1}$

${b}_{k+1}=\frac{(-1{)}^{k+1}}{(k+1)!(k+1+1)!{2}^{2(k+1)+1}}{x}^{2(k+1)+1}\phantom{\rule{0ex}{0ex}}=\frac{(-1{)}^{k+1}}{(k+1)!(k+2)!{2}^{2k+3}}{x}^{2k+3}$

It implies that

role="math" localid="1649407841088" $\underset{k\to \infty}{\mathrm{lim}}\left|\frac{{b}_{k+1}}{{b}_{k}}\right|=\underset{k\to \infty}{\mathrm{lim}}\left|\frac{\frac{(-1{)}^{k+1}}{(k+1)!(k+2)!{2}^{2k+3}}{x}^{2k+3}}{\frac{(-1{)}^{k}}{k!(k+1)!{2}^{2k+1}}{x}^{2k+1}}\right|\phantom{\rule{0ex}{0ex}}=\underset{k\to \infty}{\mathrm{lim}}\left|-\frac{1}{(k+1)(k+2)4}{x}^{2}\right|$

Calculate for *k *tending to infinity,

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

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

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