Question

{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is
J₀(x) = ∑ₖ₌₀∞ (−1)ᵏ/(2²ᵏ(k!)²) x²ᵏ
b. Find the radius and interval of convergence of the power series for J₀.

Accepted Answer

Identify the given power series for the Bessel function $$J_0(x$$)$$:

J_0(x$$) = \sum_{k=0}^\infty \frac{(-1)^k}{2^{2k} (k!)^2} x^{2k}$$ To find the radius of convergence, apply the Ratio Test to the general term

a_k$$ = \frac{(-1)^k}{2^{2k} (k!)^2} x^{2k}$$

Calculate the limit:

L$$ = \lim_{k 	o \infty} \left| \frac{a_{k+1}}{a_k} ight|$$ Substitute a$$_{k+1}$$ and a_k$ into the ratio:

$$\left| \frac{a_{k+1}}{a_k} ight| = \left| \frac{(-1)^{k+1}}{2^{2(k+1)} ((k+1)!)^2} x^{2(k+1)} \cdot \frac{2^{2k} (k!)^2}{(-1)^k x^{2k}} ight| = \left| \frac{x^2}{4} \cdot \frac{(k!)^2}{((k+1)!)^2} ight|$$ Simplify the factorial expression:

$$\frac{(k!)^2}{((k+1)!)^2} = \frac{(k!)^2}{(k+1)^2 (k!)^2} = \frac{1}{(k+1)^2}$$

So the ratio becomes:

$$\left| \frac{a_{k+1}}{a_k} ight| = \left| \frac{x^2}{4 (k+1)^2} ight|$$ Take the limit as k$$ 	o \infty$$:

L$$ = \lim_{k 	o \infty} \left| \frac{x^2}{4 (k+1)^2} ight| = 0$$

Since L = 0$ for all real x$, the series converges for all x$. Therefore, the radius of convergence is infinite, and the interval of convergence is $$(-\infty, \infty)$$.

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Key Concepts

Power Series and Radius of Convergence

Ratio Test for Convergence

Bessel Functions as Power Series