Verify my working for a math problem

[Logistics]

Could someone please verify my working out, let me know if my answer is correct & if I have written the question out properly

http://img280.echo.cx/img280/9528/solution9lx.gif

Thanks

 

[dextercioby]

It doesn't converge for every possible "x",as your "radius of convergence =infinite" might mean.

$$\sum_{k=1}^{\infty} (-1)^{k}\frac{x^{2k}}{4^{k}(k!)^{2}} =-\frac{x^{2}}{4}\ _{2}F_{1} \left(1,2,2;-\frac{x^{2}}{4}\right)$$


Daniel.

 

[heman]

To simplify
there was no need to take the x's ...take x^2=w and just take the limit of the ratio's..

As per my knowledge it seems to me R is infinite

 

[dextercioby]

Well,try x=60.How big is the number...?

Daniel.

 

[Logistics]

Thanks for the replies guys.

I have left it as R = infinity; seems the rest of the class got the same thing. So I'll just leave it at that

 

[shmoe]

It doesn't converge for every possible "x",as your "radius of convergence =infinite" might mean.

$$\sum_{k=1}^{\infty} (-1)^{k}\frac{x^{2k}}{4^{k}(k!)^{2}} =-\frac{x^{2}}{4}\ _{2}F_{1} \left(1,2,2;-\frac{x^{2}}{4}\right)$$

You've missed the k=0 term, though this doesn't affect convergence. The OP's work is fine.

For interests sake, this thing is a Bessel function of the first kind (it's a solution to the d.e. xy''+y'+xy=0).