Explain briefly the theory of recurring series, shewing that if \(2r\) terms of the series are given it can in general be continued as a recurring series of the \(r\)th order in one way only. Find the \((n+1)\)th term of the recurring series \[ -2+2x+14x^2+50x^3+\dots. \]
Explain the method of proving theorems by mathematical induction. Shew that the series \[ \frac{1}{u_0} + \frac{x}{u_0 u_1} + \frac{x^2}{u_0 u_1 u_2} + \dots + \frac{x^n}{u_0 u_1 \dots u_n} \] is identically equal to a continued fraction with \(n+1\) components, in which the first is \(\frac{1}{u_0}\), the second is \(-\frac{u_0^2 x}{u_0 x+u_1}\) and the rth is \(-\frac{u_{r-1}^2 x}{u_{r-1}x+u_r}\). Hence, or otherwise, prove that \[ \frac{1}{1-} \frac{1}{3-} \frac{4}{5-} \dots \frac{n^2}{-2n+1} = 1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n+1}. \]
Make rough drawings of the curves (i) \(y = \dfrac{x^2}{1+x^2}\); (ii) \(y = \dfrac{1-x+x^2}{1+x+x^2}\); (iii) \(y = \lim_{n\to\infty} \dfrac{x^{2n}\tan\frac{\pi x}{2} + x}{x^{2n}+1}\).
Differentiate (i) \(\dfrac{(1+x^2)^{\frac{1}{2}}+(1-x^2)^{\frac{1}{2}}}{(1+x^2)^{\frac{1}{2}}-(1-x^2)^{\frac{1}{2}}}\); (ii) \(\sin^{-1}\{\log(x^2-1)\}\). Prove that, if \[ V(u,v,w) = \begin{vmatrix} u, & v, & w \\ u', & v', & w' \\ u'', & v'', & w'' \end{vmatrix}, \] where dashes denote differentiation with regard to \(x\), then \[ V(u,v,w) = w^3 V\left(u/w, v/w, 1\right). \]
Find a formula for the radius of curvature at a point on a curve \(\phi(x,y)=0\). Prove that the equation of the evolute of the hypocycloid \(x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}\) is \[ (x+y)^{\frac{2}{3}}+(x-y)^{\frac{2}{3}} = 2a^{\frac{2}{3}}. \]
Prove the following results: \[ \int_0^\pi \frac{dx}{a+b\cos x} = \frac{\pi}{\sqrt{(a^2-b^2)}} \quad (a^2>b^2 \text{ and } a>0). \] \[ \int_{-1}^1 \frac{\sin \alpha dx}{1-2x\cos\alpha+x^2} = \frac{1}{2}\pi \quad (0<\alpha<\pi) \] \[ = -\frac{1}{2}\pi \quad (\pi<\alpha<2\pi). \] \[ \int_0^1 x^{2n-1}\log(1+x)dx = \frac{1}{2n}\left\{ \frac{1}{1\cdot2}+\frac{1}{3\cdot4}+\dots+\frac{1}{(2n-1)2n} \right\}. \]
A circle of radius \(b\) rolls on the outside of a circle of radius \(a\) and a point on the circumference of the rolling circle traces an epicycloid. Prove that the length of the arc from one cusp to the next is \(8b(a+b)/a\), and that the area between this arc and the circle is \(\pi b^2(3a+2b)/a\).
Give an account of the method of reciprocation with respect to a circle, and illustrate its use.
Give a general account of the resolution of a fraction (whose numerator and denominator are polynomials) into partial fractions. Resolve into partial fractions \(\dfrac{x^2}{(x+2)^2(x^2+1)}\).
A pole DE, inclined to the vertical, stands at D on a horizontal plane, and A, B, C are three collinear points in the plane, such that AB=\(a\), BC=\(c\). The angles DAC, DCA (\(\theta, \phi\)) and the elevations (\(\alpha, \beta, \gamma\)) of E as seen from A, B, C are measured. Shew how to determine from these measurements the inclination of the pole to the vertical and the direction in which it leans.