Show how to expand the function $$\frac{x}{(x-1)(x-2)}$$ as a power series \(a_1x + a_2x^2 + ...\). State the value of \(a_n\) and also the range of values of \(x\) for which the expansion is valid. Sum the series $$a_1x + 2a_2x^2 + 3a_3x^3 + ...,$$ $$a_1x + \frac{1}{2}a_2x^2 + \frac{1}{3}a_3x^3 + ....$$
Find the ranges of values of \(x\) for which the function \((\log x)/x\) (i) increases, (ii) decreases, as \(x\) increases. Hence determine the largest possible value of the positive constant \(k\) such that the inequalities \(0 < x < y < k\) imply that \(x^y < y^x\).
If $$I_m = \int_0^{1\pi} \sin^m x dx,$$ evaluate \(I_m\) for all positive integers \(m\). Prove that \(I_{2n-2} < I_{2n-1} < I_{2n}\), and deduce that $$\frac{2.4.6...2n}{1.3.5...(2n-1)} \cdot \frac{1}{\sqrt{n}}$$ tends to the limit \(\sqrt{\pi}\) as \(n\) tends to infinity.
Obtain Leibniz's formula for the \(n\)th derivative of the product \(u(x)v(x)\). If \(y = \frac{1}{2}(\sinh^{-1} x)^2\), prove that $$(1+x^2)y'' + xy' - 1 = 0,$$ and deduce the value of \(y^{(n)}\) (the \(n\)th derivative of \(y\)) for \(x = 0\). Obtain the Maclaurin expansion for the function.
A function defined on a plane can be expressed as \(u(r, \theta)\) or \(f(r, \theta)\), where \(r = r\cos\theta\) and \(y = r\sin\theta\). Prove that $$\frac{\partial^2 u}{\partial r^2} + \frac{\partial^2 u}{\partial \theta^2} = \frac{\partial^2 f}{\partial r^2} + \frac{1}{r^2}\frac{\partial^2 f}{\partial \theta^2}.$$ If \(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0\) and \(f = r^n \phi(\theta)\), determine the function \(\phi\).
By graphical considerations, or otherwise, show that the equation $$x = 1 + \lambda e^x$$ has real solutions if \(\lambda\) is small enough, and that one of these solutions tends to the value 1 as \(\lambda\) tends to zero. Obtain an approximate solution for this solution in the form $$x = 1 + a_1\lambda + a_2\lambda^2 + ...,$$ and determine the coefficients as far as \(a_4\).
By considering \(\int_1^2 \log x dx\) evaluate the limit, as \(n\) tends to infinity, of $$\left[\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)\left(1+\frac{3}{n}\right)...\left(1+\frac{n-1}{n}\right)\right]^{\frac{1}{n}}.$$
Prove, by substitution or otherwise, that the solution of the differential equation \(y'' + n^2y = f(x)\) with the conditions \(y(0) = y'(0) = 0\) is $$y(x) = \int_0^x \frac{1}{n}\sin(x-t)f(t)dt.$$ Solve the problem in the particular case \(f(x) = \sin nx\).
(i) Find $$\lim_{n\to\infty} \{\sqrt{n^2+n+1}-n\}.$$ (ii) Positive numbers \(x_0\) and \(y_0\) are given. \(x_1\) and \(y_1\) are the arithmetic and geometric means of \(x_0\) and \(y_0\); \(x_2\) and \(y_2\) are the arithmetic and geometric means of \(x_1\) and \(y_1\); and so on. Show that \(x_n\) and \(y_n\) tend to finite limits as \(n\) tends to infinity, and that these limits are equal.
Starting with any definition you please, establish the principal properties of the function \(\log x\), including a proof that \((\log x)/x^k\) tends to zero as \(x\) tends to infinity, for any positive number \(k\).