# What are some remarkable and interesting uses of AM-GM Inequality ? Cite and explain with problems.

There are really lot of problems on AM-GM inequality because of its elementary nature and powerful applications.

What I want is a collection of questions/problems which look very complex but get solved swiftly and powerfully through use of AM-GM Inequality.

Edit:I see that 'interesting' is subjective hence to make this question more specific please provide examples which seem interesting or remarkable to you and don't think about what's interesting to me.

• I think a good example are those optimisation exercises over the unit sphere which can be solved elegantly with mean inequalities. – Surb Dec 19 '14 at 16:39
• Since remarkable and interesting are subjective criteria, it would probably be better for you to do your own search through Questions here that involve AM-GM inequality in their Answers. That way you take responsibility for collecting the ones that fit your ideas. – hardmath Dec 19 '14 at 16:39
• An example: math.stackexchange.com/questions/839433/… – Surb Dec 19 '14 at 16:42
• Here's a search of the kind I suggested. Variations may give you better results. – hardmath Dec 19 '14 at 16:45
• No thanks are needed. I prefer to give you the tools to do your own work, believing that is how learning takes place. – hardmath Dec 19 '14 at 16:47

Lemma: Let $\{x_n\}$ be a sequence of positive real numbers. Then $$\prod_{n=1}^{\infty} (1+x_n)$$ converges if and only if $$\sum_{n=1}^{\infty} x_n$$ converges.
Proof: One direction is clear, as $\prod_{n=1}^{N} (1+x_n) > \sum_{n=1}^{N} x_n$. For the other direction, use the AM-GM inequality. $$\sqrt[N]{\prod_{n=1}^{N} (1+x_n)}\le \frac{\sum_{n=1}^{N} (1+x_n)}{N} \implies \prod_{n=1}^{N} (1+x_n) \le \left(1 + \frac{x_1 + \cdots + x_N}{N} \right)^N$$ where the right hand side converges to $\exp{\sum_{n=1}^{\infty} x_n}$.
The usual way of proving this lemma is using $1+x\le e^x$ to get that upper bound. For some reason, I find it easy to forget this inequality, so AM-GM is a good fallback.
• You can use that $t\leqslant \exp({t-1})$. – Pedro Tamaroff Dec 19 '14 at 17:25