Problem with Taylor's Theorem

EDIT: tHE QUESTION CAN BE SEEN HERE:

IMG_2571.jpg - Uploader v6

Hey guys, im having a problem with using taylor's theorem. Well actually i dont know where to start. I need to use taylors theorem to find the value of the BESSEL FUNCTION OF ORDER 1 of J(1->5)(0) where J(x) Σ k=0-->∞

(-1)^k x^2k+1/ k!(k+1)! 2^2k+1

(scroll down for real translation)

So i guess i need to find the 1st 5 terms of this. But the only kind of taylors we have been taught is starting with a function and deriving. This starts with an infinite series so i dont know how to approach this. Sorry its a little bad looking so let me know if you need any clarification. Thanks!

Chris