Taylor Series / Power Series

Alright, I'm just a little confused on what my work is asking me. The problem says "Use the definition to find the Taylor series, centered at c, for the function."

The problem is:

$\displaystyle \cos(x), c = \pi/4$

So I have an answer book that has the answer for this problem. I understand Taylor series / power series, but I'm just a bit confused on what is going on.

The answer book goes on to do a few derivatives, plugs in the c, etc. Then it forms the sum which is:

$\displaystyle \frac{\sqrt2}{2}\sum{\frac{(-1)^{n(n+1)/2}(x-\frac{\pi}{4})^n}{n!}}$

Now I understand this is in the form of:

$\displaystyle \sum{\frac{f^n(c)(x-c)^n}{n!}}$

Yet... Why does it not follow the "basic list" of power series? Which has:

$\displaystyle \cos(x) = \sum{\frac{(-1)^nx^{2n}}{(2n)!}}$

Why is there only n powers in the answer, and no 2n to be found, etc.? Can someone please explain? Maybe I am mixing two concepts...