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Let $(I_n)_{n\ge2}$, $I_n=\int_0^{\pi/4}{\tan^n x \, dx}$

Find $\lim_{n\to\infty}\int_0^{\pi/4}{\tan^n x \, dx}$.
$I_n$ is a sequence of (definite) integrals. I don't know any other way to calculate a limit of such sequence than using the Squeeze Theorem.

I can't find two bounds for $I_n$ whose limits as $n\to\infty$ are equal. First thing I tried was considering the double inequality $0 \le x \le \pi/4$ and rewriting it I got $0 \le I_n \le \pi/4$. I realize this is only true for $n \ge 0$ which is not the case in this exercise because $n \ge 2$. $I_2 = 1 - \pi/4$ so this is the upper bound of the $I_n$, then the double inequality becomes: $$0 \le I_n \le 1 - \frac{\pi}{4}$$
I can't apply Squeeze Theorem because limit of left term is not equal to the right term and this is often the case with a lot of exercises of this kind that I attempt. I realize that I need to find two functions (of $n$) that can substitute the bounds for $I_n$, but I just have no clue what those would be, please give me a hint.

Radu Gabriel
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1 Answers1

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For $x \in [0, \tan \frac{\pi}{4})$ you have $0 \le \tan x <1$. Hence $ x \mapsto \tan x$ converges pointwise to zero on this interval. As the map is also bounded, the sequence $\{I_n\}$ converges to zero according to Lebesgue Dominated Convergence theorem.

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