I know this represents the cos(x) function, but what does it mean that the summation starts with 1 and ends with 21?
This question is for my review sheet for an exam I have. Also, if I were to write the last three terms, does that mean they would be $x^6/6!, x^8/8!$, and $x^{10}/10!$ ?
If you truncate a Taylor series to a polynomial, you get something that's a good approximation of the function ... if you're close enough. For larger $x$, the fact that it's just a polynomial takes over, and it separates from the function completely to do its own thing. Since it's a polynomial, that thing is to go to $\pm\infty$ as $x\to\infty$ or $x\to -\infty$. Which sign? That depends on the sign of the coefficient of the largest-degree term and whether that degree is even or odd.