The Squeeze Theorem And Trigonometric Inequalities

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When dealing with sequences and series, we often use inequalities to deal with trig functions. As an example, consider the sequence

$\displaystyle a_i = \frac{\sin i}{i^2}$

Does this sequence converge? It’s hard to tell, because $\sin i$ behaves strangely for integer values of $i$ . Sometimes it’s positive, sometimes it’s negative, and there’s not much of a pattern to it all. However, one thing we know (because $\sin x$ is a bounded function) is that the inequalities

$\displaystyle -1\leq \sin i\leq 1$

is true for all $i$ . Dividing everything by $i^2$ , we get the inequalities

$\displaystyle \frac{-1}{i^2}\leq\frac{\sin i}{i^2}\leq\frac{1}{i^2}$

Since both the sequences $-1/i^2$ and $1/i^2$ converge to zero, our sequence $a_i$ does, too, by the squeeze theorem.

This type of reasoning is very common when you see sines and cosines in a sequence.

Sometimes these types of inequalities are useful for non-trig functions, too. For example, the function $e^{-x}$ is bounded on the interval $[1, \infty)$ . What are the bounds? Can you use this fact to show that the sequence