What is the maximum value of $\frac{x^{100}}{1+x+x^2+\ldots+x^{200}}$?

Question

If $x$ is positive, what is the maximum value of this expression:

$$\frac{x^{100}}{1+x+x^2+\ldots+x^{200}}$$

This question is from a book of problems on sequence and series under the section on AM-GM-HM inequality.

This is what I have tried:

The denominator is a geometric series whose sum is

$$\frac{1-x^{201}}{1-x}$$

The fraction now becomes

$$\frac{x^{100}(1-x)}{1-x^{201}}$$

I can imagine that solving this problem will require taking the AM/GM/HM of some expressions of $x$ and applying the AM-GM-HM inequality.

That means the above fractions should themselves be one of GM or HM (whose maximum value will be given by the corresponding AM and GM respectively).

I can't see such means from looking at the fraction. Can someone help me here?

Answer 1

The denominator (in the original form) is a multiple of an arithmetic mean - a sum of $201$ terms is $201$ times their average. So then, depending on taste, you can either apply AM-GM to the denominator or GM-HM to the whole thing.

Answer 2

For $x>0$, we have from the AM-GM inequality

$$\begin{align} \sum_{n=0}^{200}x^n&\ge 201 \sqrt[201]{\prod_{n=0}^{200}x^n}\\\\ &=201 \sqrt[201]{x^{20100}}\\\\ &=201x^{100} \end{align}$$

Hence, we see that

$$\frac{x^{100}}{\sum_{n=0}^{200}x^n}\le \frac1{201}$$

Answer 3

You can instead minimise the reciprocal of your quantity, viz., $$\frac{1+x+x^2+\cdots+x^{200}}{x^{100}}=x^{-100}+x^{-99}+\cdots+x^{99}+x^{100}.$$ One only needs the two-variable AM/GM inequality to do this, just in the form $y+y^{-1}\ge2$ for $y>0$, for $$x^{-100}+x^{-99}+\cdots+x^{99}+x^{100}=1+\sum_{n=1}^{100}(x^n+x^{-n}) \ge201$$ with equality if $x=1$.