For the first, notice that all even-numbered terms cancel since . For the second, you're just adding the term corresponding to , I don't really know what you did to get to the other expression.
For the truncated form of the Taylor series, the Nth Taylor polynomial is given in our notes as
But then later, he defines for
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
but is it not: