Prove that the series $$\frac{1}{1} + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \ldots$$ is divergent. Prove also the series $$\frac{1}{1} + \frac{1}{4} + \ldots + \frac{1}{81} + \frac{1}{100} + \ldots + \frac{1}{8^2} + \frac{1}{100} + \ldots,$$ derived from the first series by the omission of all terms whose denominators contain the digit 9, is convergent.
Verify that the differential equation $$x^2 y'' + [(n + \frac{1}{2})x + \frac{1}{2}](1-x^2)]y = 0,$$ where \(n\) is a positive integer, has the solution $$y = x^{\frac{1}{2}} e^{-\frac{1}{2}x} L_n(x),$$ where \(L_n(x)\) is the polynomial of degree \(n\) given by $$L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n}(x^n e^{-x}).$$
Prove that \((\sin x)/x\) is a decreasing function of \(x\) for \(0 < x < \frac{1}{2}\pi\). Assuming that \(F(x) \geq 0\) when \(a \leq x \leq b\) implies that \(\int_a^b F(x)dx \geq 0,\) prove that, if \(m \leq f(x) \leq M\) when \(a \leq x \leq b\), \(m(b-a) \leq \int_a^b f(x)dx \leq M(b-a),\) and deduce that \(I = \int_0^{\pi/3} \frac{\sin x}{x} dx\) lies between \(0.866\) and \(1.048\). Prove further that, if also \(\phi(x) > 0\) when \(a \leq x \leq b\), \(m \int_a^b \phi(x)dx \leq \int_a^b f(x)\phi(x)dx \leq M \int_a^b \phi(x)dx,\) and by making the substitution \(x = 2y\) prove that \(I\) lies between \(0.955\) and \(1\).
The horizontal carriageway of a suspension bridge is suspended from a chain of \(2n+1\) light links by \(2n\) light vertical rods at a constant distance \(a\) apart (so that the links carry in length). The ends of the chain are fixed at points at the same level at a distance \(2na\) apart. If the tension in the \(k\)th link (\(k = 0, 1, 2, \ldots, n-1, n\)) is \(T_k\) and the lengths of the rods attached to its ends are \(y_k\) and \(y_{k+1}\), show that \(y_{k+2} - 2y_{k+1} + y_k = \frac{aW}{T_0},\) and find \(y_k\) and \(T_k\) in terms of \(y_0\) and \(T_0\).
The polar coordinates of a moving particle are \((r, \theta)\). Prove that the radial and transverse components of its acceleration are \(\ddot{r} - r\dot{\theta}^2\) and \(2\dot{r}\dot{\theta} + r\ddot{\theta}\). A particle moves under the action of a force directed towards the origin and of magnitude \(\mu\) per unit mass (\(\mu\) constant). Establish the equations of conservation of energy and moment of momentum: \(\frac{1}{2}(\dot{r}^2 + r^2\dot{\theta}^2) - \frac{\mu}{r} = E, \quad r^2\dot{\theta} = h,\) and prove that the differential equation of the orbit is \(\frac{d^2u}{d\theta^2} + \left(u - \frac{\mu}{h^2}\right) = 0,\) where \(u = 1/r\). If the particle is initially at a point \(A\) at a distance \(c\) from the origin \(O\), and its velocity is at right angles to \(OA\) and of magnitude \(V\), find the conditions that the orbit shall be (i) an ellipse, (ii) an ellipse with its centre between \(O\) and \(A\).
A circular hoop of radius \(a\) rolls along the ground with velocity \(U\). It strikes a horizontal bar fixed at height \(2a/5\), rotates about the bar until it touches the ground again, and then rolls along the ground with velocity \(V\). If the hoop does not slip on or rebound from the bar or the ground, show that \(10ga < 16U^2 < 16ga\) and \(25V = 16U.\)
Defining the coefficient of correlation between two variables \(x\) and \(y\) as \(\rho = \frac{E[(x-Ex)(y-Ey)]}{\sqrt{E[(x-Ex)^2]} \cdot \sqrt{E[(y-Ey)^2]}},\) where \(E\) denotes the expectation value, prove
There are 50,000 shares in a lottery with 1000 prizes. If a syndicate buys 100 shares, write down an approximate expression for, and evaluate approximately, the chance that it wins four or more prizes. Find also the variance of the number of prizes that may be expected. [The approximation \((1 + 1/n)^{xn} = e^x\) for large \(n\) may be used.]
If \(\alpha\) is a complex fifth root of unity, prove that \(\alpha - \alpha^4\) is a root of the equation $$\alpha^4 + 5\alpha^2 + 5 = 0.$$ Express the other roots of this equation in terms of \(\alpha\).
If \(u = x + y\), \(v = xy\), and \(x^n + y^n = 1\), find the degree in \(v\) of the algebraic relation between \(u\) and \(v\). If \(n = 5\), prove that $$5(x + y)(1 - x)(1 - y)(1 - x - y + x^2 + xy + y^2) = (x + y - 1)^5.$$