Deriving a Taylor Series for Sinx: Is it the Same as a Power Series?

[nameVoid]

Is it correct to take the derivative of a taylor series the same as you would for a power series ie:
$$sinx=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!}$$
$$\frac{d}{dx}(sinx)=cosx=\sum_{n=1}^{\infty}(-1)^n(2n+1)\frac{x^{2n}}{(2n+1)!}$$
it seems as if it wouldn't be
$$cosx=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}$$

 

[lanedance]

have a look at your limits of summation

 

[g_edgar]

Also, is this true?

$$\frac{(2n+1)}{(2n+1)!} = \frac{1}{(2n)!}$$

for example

$$\frac{7}{7!} = \frac{1}{6!}$$

 

[Fightfish]

Yup, that is true, you can prove it easily by factoring out (2n+1) from (2n+1)!