I will truly appreciate your help about this particular question!
I am wondering why the "the sum of the first n + 1 terms of the power series of the exponential function gives negative numbers"
Exponential function - Wikipedia, the free encyclopedia - the figure on the right
I am also not so good to understand the exponential function itself... but I hope someone could help me!
can take on negative values although the exponential function is always positive.
The reason is that the series only gives the exponential function in the limit. If you truncate the series and only take finitely many terms, you only get an approximation to , and actually the approximation is only a good fit for values near zero. If you get far away from zero the approximation is poor, and you can get negative values. The more terms in the series you take, the better the approximation gets, and you can get further away from zero before you get in trouble. That's what the animated graph on Wikipedia is trying to show.