Unorthodox Inequality

Let $n$ be a positive integer and $x$ be a positive real. Prove that $\dfrac{n(nx+1)}{x+1} \le \dfrac{(x+1)^{2n}-1}{(x+1)^2-1}$ and find equality case.

Hint (zoom in to read):

$\tiny _{\text{The inequality is still true even when loosening the restriction of }x\text{ to }x > -1}$

Bigger hint: link

#Algebra #Inequality #EqualityCase #PowerSeries #Proof

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

x>0 not x>=0 because if x=0 you get a non defined operation: 0/0 Nice problem ;)

Carlos David Nexans - 6 years, 11 months ago

True. However, if you looked at the first hint, it can even be looser.

Has anyone solved this problem yet? I have a pretty big hint as my second hint...

Daniel Liu - 6 years, 10 months ago

Nah I ended up at $1 \geq 2$ and exploded the universe.

Samuraiwarm Tsunayoshi - 6 years, 7 months ago

Final hint: multiply $x+1$ on both sides. Does the format of the resulting product resemble a known formula?

Daniel Liu - 6 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$