Forever $\dfrac{\pi}{2}$ ??

Looks like only three guys answered this, I thought that Borwein integrals are very famous.

You're really awesome Mr. Haussmann! Two thumbs up!! d(^_^)b

Consider the following sequence: $$ $\begin{aligned} U_1&=\int_0^\infty\frac{\sin x}{x}dx=\frac{\pi}{2},\\ U_2&=\int_0^\infty\frac{\sin x}{x}\cdot\frac{\sin \left(\frac{x}{3}\right)}{\left(\frac{x}{3}\right)}dx=\frac{\pi}{2},\\ U_3&=\int_0^\infty\frac{\sin x}{x}\cdot\frac{\sin \left(\frac{x}{3}\right)}{\left(\frac{x}{3}\right)}\cdot\frac{\sin \left(\frac{x}{5}\right)}{\left(\frac{x}{5}\right)}dx=\frac{\pi}{2},\\ U_4&=\int_0^\infty\frac{\sin x}{x}\cdot\frac{\sin \left(\frac{x}{3}\right)}{\left(\frac{x}{3}\right)}\cdot\frac{\sin \left(\frac{x}{5}\right)}{\left(\frac{x}{5}\right)}\cdot\frac{\sin \left(\frac{x}{7}\right)}{\left(\frac{x}{7}\right)}dx=\frac{\pi}{2},\\ U_5&=\int_0^\infty\prod_{n=1}^5\frac{\sin \left(\frac{x}{2n-1}\right)}{\left(\frac{x}{2n-1}\right)}dx=\frac{\pi}{2},\\ U_6&=\int_0^\infty\prod_{n=1}^6\frac{\sin \left(\frac{x}{2n-1}\right)}{\left(\frac{x}{2n-1}\right)}dx=\frac{\pi}{2},\\ U_7&=\int_0^\infty\prod_{n=1}^7\frac{\sin \left(\frac{x}{2n-1}\right)}{\left(\frac{x}{2n-1}\right)}dx=\frac{\pi}{2},\\ \color{#D61F06}{\text{AND}}\\ U_8&=\int_0^\infty\prod_{n=1}^8\frac{\sin \left(\frac{x}{2n-1}\right)}{\left(\frac{x}{2n-1}\right)}dx=\color{#3D99F6}{\frac{467807924713440738696537864469}{935615849440640907310521750000}\pi}.\\ \end{aligned}$ $$ In mathematics, these integrals are called Borwein integrals . This fact is discovered by David and Jonathan Borwein. In general similar integrals have value $\dfrac{\pi}{2}$ whenever the numbers $3, 5, 7\cdots$ are replaced by positive real numbers such that the sum of their reciprocals is less than $1$ . In the example above, $\frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{13} < 1$ , but $\frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{15} > 1$ .

Fun fact : After this was discovered by David and Jonathan Borwein, Jonathan verified that the computer algebra software package Maple reports the correct values for all these integrals — and then, as a practical joke, reported this to Maple as a “bug” in the software. Maple computer scientist Jacque Carette reports that “I must have spent three days on this project before I figured out that Jon had tricked me”. $$

$\text{\# }\mathbb{Q.E.D.}\text{ \#}$