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?
Suppose that has the property that for any even we have no real roots. We can easy show that has property that is has exactly one real root. Note that where is even, and so the derivative is non-vanishing for , thus it is an increasing odd polynomial, thus, it has exactly one root.
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.
Are you familar with Taylor's theorem because that is all you need to know.