GATE 2020 MA – Question 31

Real Analysis · Sequences and Series of Functions · 1 mark · Numerical answer

For n ≥ 2, define $x_n=(2\pi)^{-1}\int_0^{\pi/2}\tan^{1/n}t\,dt$. Find its limit, to two decimal places.

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Correct answer: 0.25 to 0.25

Explanation

For 0 < t < π/2, $\tan^{1/n}t\to1$. For n ≥ 2 these functions are bounded by $1+\sqrt{\tan t}$, which is integrable on this interval: near π/2 its singularity is of order $(\pi/2-t)^{-1/2}$. Dominated convergence therefore gives limit $(2\pi)^{-1}(\pi/2)=1/4=0.25$.

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