Taylor Series and shifting indices

For the truncated form of the Taylor series, the Nth Taylor polynomial is given in our notes as

$\displaystyle T_{N}(x) = \sum_{n = 0}^N \frac{f^{(n)}(0)}{n!} \cdot x^{n}$

But then later, he defines $\displaystyle T_{3}(x)$ for $\displaystyle f(x)=\sin x $

as $\displaystyle x-\frac{x^{3}}{6}$

Should it not be expanded to three terms, or is the notation signifying that it should be expanded until n, rather than N, reaches 3?

Also, I have a question about shifting indices. Our notes give this equality

$\displaystyle \sum_{n = 1}^\infty \frac{(-1)^{n}4^{n}}{n!} = \sum_{n = 1}^\infty \frac{(-4)^{n}}{n!} = \sum_{n = 0}^\infty \frac{(-4)^{n}}{n!} - 1 = e^{-4} - 1 , $

but is it not:

$\displaystyle \sum_{n = 1}^\infty \frac{(-4)^{n}}{n!} = \sum_{n = 0}^\infty \frac{(-4)^{n}}{n!} - \frac{4}{n+1}?$

Thanks :)