1. Bessel Function convergence

EDIT: Mods can delete this as they see fit, as I made a clearer and more urgent post!

the function y is defined as a the expansion of a series (which i know to be the bessel function, though i only know the expansion terms):

y = 1 - (2^-2)*x^2 + (2^-2)*(4^-2)*x^4 - (2^-2)*(4^-2)*(6^-2)*x^6 + ...

thus i believe the power series can be defined as:

y = sum((-1/4)^n*(n!)^2*x^(2*n), n = 0 -> inf)

my goal is to show this series converges for any value of x.

i can't seem to use the radius of convergence approach, because that requires taking the limit of the ratio of successive terms, which seems to be zero, and by my book's definition of this radius, abs(x) < R converges, abs(x) > R diverges, so the series would certainly diverge.

i'm not the most familiar with showing convergence in a series, so any help is appreciated.

2. Originally Posted by primasapere
the function y is defined as a the expansion of a series (which i know to be the bessel function, though i only know the expansion terms):

y = 1 - (2^-2)*x^2 + (2^-2)*(4^-2)*x^4 - (2^-2)*(4^-2)*(6^-2)*x^6 + ...

thus i believe the power series can be defined as:

y = sum((-1/4)^n*(n!)^2*x^(2*n), n = 0 -> inf)

my goal is to show this series converges for any value of x.

i can't seem to use the radius of convergence approach, because that requires taking the limit of the ratio of successive terms, which seems to be zero, and by my book's definition of this radius, abs(x) < R converges, abs(x) > R diverges, so the series would certainly diverge.

i'm not the most familiar with showing convergence in a series, so any help is appreciated.
Let us consider the simplest case of the Bessel function (I think it is called Bessel functions of the first kind).

I am not familar with Bessel functions, however I know they are derived from the differencial equation,
$x^2y''+xy'+x^2y=0$
We realize that, $x=0$ is a regular singular point.
Then, we use a theorem about regular singular linear second order homogenous differencial equations. You have to look at the radii of convergence of the two analytic functions $p(x), q(x)$ in,
$x^2y''+xp(x)y'+x^2q(x)y''=0$
Then an analytic solution exists and its radii of convergence is as least as large as of $p(x),q(x)$.

Basically what I am trying to say is that there is a theorem that gaurenttes a solution with a certain radii of convergence.
In the case of Bessel functions,
$x^2y''+xy'+x^2y'=0$
The two functions,
$p(x)=1, q(x)=1$ are analytic with radii of convergence the entire number line. Thus there exists an analytic solution (Bessel function) whose radii of convergence is at least as large as the other two (that is the entire number line).

Does that help at all? This is a subject I am not well versed, but I think knowing that theorem might help.

3. that's... kind of a scary explanation! i sort of understand the idea, though i don't really understand the theorem used.

i don't think i'm supposed to know that i'm dealing with a bessel function. or, rather, i don't think i'm supposed to use it to arrive at my answer.

is there any other way to see that the series converges for any value of x that's based solely on the terms of the expansion?

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convergence of bessel function

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