Answer to Question #304168 in Calculus for itz_perfect

Question

The function J1 defined by

J1(x) = ∑((((−1)^n).x^(2n+1))/(n! (n + 1)! 2)).2n+1

n=0

,

is called the Bessel function of order 1.

a) Find its domain.

b) Graph the first several partial sums on a common screen using programming software.

Answer 1

a) the domain of the function is the interval of convergence of power series. Let $a_n=\frac{(-1)^{n}x^{2n+1}}{n!(n+1)!2^{2n+1}}$ and find the $lim_{n\to\infin}|\frac{a_{n+1}}{a_n}|$

$|\frac{\frac{(-1)^{n+1}x^{2(n+1)+1}}{(n+1)!((n+1)+1)!2^{2(n+1)+1}}}{\frac{(-1)^{n}x^{2n+1}}{n!(n+1)!2^{2n+1}}}|=|\frac{x^{2n+3}}{(n+1)!(n+2)!2^{2n+3}}\times\frac{n!(n+1)!2^{2n+1}}{x^{2n+1}}|$

$=|\frac{x^2}{(n+1)n!(n+1)(n+2)!2^2}\times n!(n+1)!|$

$=|\frac{x^2}{4(n+1)(n+2)}|$

$\to0\ as\ n\ \to\infin$

By ratio test, Since $lim_{n\to\infin|\frac{a_{n+1}}{a_{n}}|}=0\le1\ \forall\ x$ , The given series converges ,With this ,write write the interval which is exactly the domain of the function

$(-\infin,\infin)$

b)