Find function is analytic on R and has Maclaurin expansion

[HF08]

Prove the function is analytic on R and find its Maclaurin expansion.

(a) e$$^{x}$$cosx


Well, I did some work. I can show that

e$$^{x}$$=$$\sum$$x$$^{k}$$/k!

cosx=$$\sum$$((-1)$$^{k}$$x$$^{2k}$$)/(2k)!

are analytic and have the above Maclaurin expression.

My problem is multiplying these two expressions together, finding the Maclaurin
expression and proving it is analytic.

I would be grateful for all your help. I think the hardest part will be showing how
it is analytic.

Thanks,
HF08

 

[EnumaElish]

If you were to multiply the two series like you would two polynomials, wouldn't that give you a power series representation of exp(x)cos(x)?

Here is a link that may be useful: http://en.wikipedia.org/wiki/Cauchy_product

 

[HF08]

I know the Maclaurin Series now :-)

e$$^{x}$$cos x = $$\sum$$$$\sum$$[(-1)$$^{j}$$]/((2j)!(k-2j)1))

Whew! That was a lot of work for me, but I got it. Thanks. The result is correct. So how do I show this is analytic? I'm stuck, please, please help me.

Thanks,
HF08