Question

. (5pont)Thedale integraltegralsovertherduis an improper integ da dy is an improper integral that could be defined as the limit of double integrals over the rectangle [0,t] x [0, t] as t-1. But if we expand the integrand as a geometric series, we can express the integral as the sum of an infinite series. Show that Tl 2. (5 points) Leonhard Euler was able to find the exact sum of the series in the previous problem. In 1736 he proved that 1 2 n=1 In this problem we ask you to prove this fact by evaluating the double integral in the previous problem. Start by making the change of variables u+v V2 L-U ェ=- This gives a rotation about the origin through an angle t/4. You will need to sketch the corresponding region in the uw-plane.

Answer 1

Solution Ctiven dx dy le 6 2 n 2 6 n의 Cs Scanned with CamScanner

2 Co h2. l-xy Cs Scanned with CamScanner