Problem related to Real Analysis

Question

I found a question, I don't understand How to do this.

The question is

Prove that the equation $1-x+\frac{x^2}{2}-\frac{x^3}{3}+....+(-1)^n\frac{x^n}{n}=0$ has one real root if $n$ is odd.

I found this question on a book containing many questions on real analysis. I don't know which chapter it is belonging from.

Answer 1

I'll denote the polynomial by $f_n(x)$.

• Show $f_n$ has at least one real root if $n$ is odd. (This part is not difficult, and I actually would have hoped to see it in the question, so I'm completely leaving it to you.)
• Suppose $n$ is odd and $f_n$ has at least two real roots. Then by Rolle's theorem, $f_n'$ has a real root. Now look at $-f_n'(-x)$. This is a special kind of polynomial; you should be able to show it has no real roots. If you find that you can't, look back over some material on elementary complex numbers.