Problems

Assuming that \(x\{\log(1+x)\}^{-1}\) can be expanded in ascending powers of \(x\), find the first four terms in the expansion. Hence show that a capital sum accumulating at compound interest at \(r\) per cent. per annum will be increased tenfold after \(\left(\frac{230.26}{r}+1.15\right)\) years.

Sum the series:

1. [(i)] \(\displaystyle\frac{2^3}{1!} + \frac{3^3}{2!} + \frac{4^3}{3!} + \dots\) to infinity,
2. [(ii)] \(\displaystyle\frac{1}{1.2.3} + \frac{1}{2.3.4} + \frac{1}{3.4.5} + \dots\) to \(n\) terms.

Eliminate \(\alpha, \beta, \gamma\) from the equations: \begin{align*} \cos\alpha+\cos\beta+\cos\gamma &= l, \\ \sin\alpha+\sin\beta+\sin\gamma &= m, \\ \cos2\alpha+\cos2\beta+\cos2\gamma &= p, \\ \sin2\alpha+\sin2\beta+\sin2\gamma &= q. \end{align*}

Prove that if \(n\) is a positive integer, \[ \cos nx - \cos n\theta = 2^{n-1}\prod_{r=0}^{n-1}\left\{\cos x - \cos\left(\theta+\frac{2r\pi}{n}\right)\right\}. \] Deduce a product for \(\sin n\theta\). Also show that \[ \cos\frac{\pi}{n}\cos\frac{2\pi}{n}\dots\cos\frac{(2n-1)\pi}{n} = \frac{(-1)^n-1}{2^{2n-1}}. \]

Investigate the maxima and minima of the function \((x+1)^5/(x^5+1)\) and trace its graph. Prove that the equation \((x+1)^5=m(x^5+1)\) has three real roots if \(016\).

Prove Taylor's Theorem, obtaining a form for the remainder after \(n\) terms. Apply the theorem to obtain an infinite series for \(\log(1+x)\) valid when \(-1

Explain the term 'point of inflexion' of a plane curve, and prove that if \(y=f(x)\) has a point of inflexion whose abscissa is \(x_0\), then \(f''(x_0)=0\). The graph of a polynomial of the fourth degree in \(x\) touches the \(x\)-axis at \((a,0)\) and has a point of inflexion at \((-a,0)\). Prove that the graph passes through \((-2a,0)\) and that it has a second point of inflexion whose abscissa is \(a/2\).

Prove that for a plane curve \(\displaystyle p=r\frac{dr}{dp}\). Prove that the radius of curvature of \(r^n=a^n\cos n\theta\) is \(a^n/(n+1)r^{n-1}\) and find the \((p,r)\) equation of the evolute of the curve.

Integrate: \[ \int\frac{dx}{(a^2+x^2)^{3/2}}, \quad \int\frac{dx}{x\sqrt{1+x+x^2}}, \quad \int\frac{dx}{a+b\cos x}. \] Find \(\displaystyle\int_0^\infty e^{-ax}\sin^nx\,dx\) by the use of a formula of reduction.

Show that the area of the surface of the spheroid formed by revolving the ellipse \(\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) about the axis of \(y\) is \(2\pi a^2\left[1+\frac{1-e^2}{e}\tanh^{-1}e\right]\), where \(e\) is the eccentricity of the ellipse.