How do I show that the function
fn(x) = 1 + x + (x^2)/2! + ... + (x^n)/n!
has no real roots when n is an even number,
and has one real root when n is an odd number?
There is an elementary way to prove that for even, which will complete what you want to show. Using this we can prove this. Let then if is even. This mean .
I will try to come up with an approximating solution to this problem also, since I like putting bounds on functions.
Here is an approximation-type proof. First if then certainly and if then . Thus, if is safe to assume that . Let by Taylor's theorem it means for any we have where because is a Taylor degree polynomial. Thus, . But since is even and . Thus, . This means, for all and even.