Is it true that $$x^{2k} - x^{2k-1} + x^{2k-2} + .... + x^2 - x \geq 0$$ for any $x$ real, and $k\geq 3$ a positive integer? It seems to be true for $k\geq 3$, but it is not true for k=2. I thought about grouping the terms two by two but I didn't have much success.
Edit: Nevermind it obviously fails for x in $(0,1)$. Ignore the question.
Edit 2: Actually, it seems that $$x^{2k} - x^{2k-1} + x^{2k-2} + .... + x^2 - x + 1 \geq 0$$ for all positive integers $k$. Any ideas on this one?
Hint: Unless $x=-1$, your expression is equal to$$\frac{x^{2k+1}-x}{x+1}.$$