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As in the title, how can I prove $$ \frac{\arctan(x)}{x}\geq\frac{1}{2} $$ for $x\in(0,1]$?

I think I can say:

$$\frac{\arctan(x)}{x}$$

is monotonically decreasing in the interval, so its value is greater than the value in $1$, which is $\frac{\pi}{4}$, greater than $\frac{1}{2}$.

There exists some elegant proof of this simple inequality, maybe using series expansions?

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2 Answers 2

up vote 2 down vote accepted

When $-1\le x\le 1$, we have $$\arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\cdots,$$ and therefore if $-1\le x\le 1$ and $x\ne 0$, $$\frac{\arctan x}{x}=1-\frac{x^2}{3}+\frac{x^4}{5}-\frac{x^6}{7}+\cdots. \tag{$1$}$$ The series $(1)$ is an alternating series, and therefore if we truncate just after the term $-\frac{x^2}{3}$ the error term is positive (and less than the first "neglected" term $\frac{x^4}{5}$). It follows that $$\frac{\arctan x}{x}\gt 1-\frac{x^2}{3}\ge \frac{2}{3}.$$

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Use that $\sin(x) <x$ for $x > 0$ (which is clear from a picture of the unit circle) and so $$\frac{\arctan(x)}{x} >\frac{\sin(\arctan(x))}{x} = \frac{1}{\sqrt{1+x^2}}$$ which seems good enough an approximation on $(0,1]$.

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+1. This is very neat. –  user17762 Jan 4 '13 at 16:46

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