Prove that $\int_{-\infty}^\infty P_n(x) \, dx = \pi /n$

Question

Let $P_n(x) = \frac{n}{1+n^2x^2}$. Prove that for every $n\in\mathbb{N}$ $$\int_{-\infty}^\infty P_n(x) \, dx = \frac{\pi}{n}$$

And for every $\delta > 0$:

$$\lim_{n\to\infty} \int_\delta^\infty P_n(x) \, dx = \lim_{n\to\infty} \int_{-\infty}^{-\delta} P_n(x) = 0$$

This question is Fourier-series-oriented and maybe has something with Fejér's theorem

I'd be glad to get a guidance.

Thanks.

Answer 1

Hint:

$$\int\frac1{1+(nx)^2}dx=\frac1n\int\frac{n\;dx}{1+(nx)^2}=\frac1n\arctan nx+C$$

BTW, the above shows that apparently you forgot a factor $\;\frac1n\;$ in the result of your integral

Addition: for every $\;\delta >0\;$ :

$$\int_\delta^\infty P_n(x) dx=\lim_{b\to\infty}\left.\frac1n\arctan nx\right|_\delta^b=\frac1n\left(\frac\pi2-\arctan\delta\right)\xrightarrow[\delta\to 0]{}\frac\pi{2n}$$

and now you can take the limit when $\;n\to\infty\;$ and get zero indeed.

Answer 2

Recall from calculus that $$ \frac{d}{d\theta} \arctan \theta = \frac{1}{1+\theta^2}. $$

Therefore $$ \frac{d}{d\theta} \left(\frac{1}{n}\arctan n\theta \right)= \frac{1}{1+n^2\theta^2}. $$ The fundamental theorem of calculus then informs us that for any $t_1$ and $t_2$ $$ \int_{t_1}^{t_2}\frac{1}{1+n^2\theta^2} \; d \theta = \frac{1}{n}(\arctan n t_1 - \arctan n t_2). $$ Note that $\lim_{\theta \rightarrow \pm \infty} \arctan \theta = \pm \pi/2$. So upon taking the limits $t_1 \rightarrow - \infty$ and $t_2 \rightarrow \infty$, we find that that $$ \int_{- \infty}^{\infty}\frac{1}{1+n^2\theta^2} = \frac{1}{n}(\pi/2 - (-\pi/2)) = \frac{\pi}{n}. $$

Using the above formula for the integral on any interval $[t_1,t_2]$ we may also readily compute the two latter two integrals. These will be equal to $$ \frac{\pi}{2n} - \frac{1}{n}\delta $$ and $$ - \frac{1}{n}\delta - \frac{\pi}{2n}, $$ respectively. Taking $n \rightarrow \infty$ then gives the desired result.