Lecture 29 1. Show the result found by Jacob Bernoulli: 1+\frac{1}{2^{2}}+\frac{1}{3^2}+\ldots+\frac{1}{n^{2}}+\ldots\leq2 2. Jacob Bernoulli proposed to determine the shape of the catenary, the curve assumed bya flexible but inelastic cord

hung freely between two fixed points. Galileo had thoughtthat this curve was a parabola. Jacob himself was unable to solve the problem, butin the Acta eruditorum for June 1691 there were solutions by Leibniz, Huygens andJohann Bernoulli. Johann was very proud that he had surpassed his old brother.Johann reduced the problem into a differential equation d x=\frac{\omega d y}{\sqrt{v^{2}-\alpha^{3}}} ^^20Show^^20that^^20x=a\ln \mleft(y+\sqrt{\left.y^{2}-\alpha^{3}\right)}\mright.^^20from^^20Calculus.^^20