Problems

Prove that if \(x^2<1\), \[ \frac{\sin\theta}{1-2x\cos\theta+x^2} = \sin\theta+x\sin 2\theta+x^2\sin 3\theta+\dots. \] Express \(\frac{\sin n\theta}{\sin\theta}\) in a series of descending powers of \(\cos\theta\).

Sum to \(n\) terms the series whose \(r\)th term is

1. [(i)] \(\cos\{\alpha+(r-1)\beta\}\),
2. [(ii)] \(\cos r\phi \sec^r\phi\),
3. [(iii)] \(\tan^{-1}\frac{\sin\alpha}{2r(r-\cos\alpha)}\).

Define the upper and lower limits of a function of an integral variable. If \(f(n)n_0\), and \(f(n)>A\) for an infinite number of values of \(n\), prove that \(\varlimsup f(n)\) exists, and that \(A \le \varlimsup f(n) \le B\). If \(G(y)\) is the upper bound of \(\phi(x)\) in \(a

Prove that a continuous function of one variable is bounded in any interval in which it is continuous. Define a continuous function of any number of variables, and extend the proposition to such a function. Discuss the continuity of \((x-y)/(x+y)\) in a rectangle which contains the origin.

Prove that under certain stated conditions the equation \(f(x,y)=0\) determines \(y\) as a unique continuous function of \(x\) in a certain \(x\)-neighbourhood. Prove that the equation \(x^3+y^3=3axy\) has a continuous solution in \(y\) as a function of \(x\) for real values of \(x\) whose modulus exceeds a certain value, and that \[ x+y = -a + a^3/x^2 + O(1/x^3), \] as \(x\to+\infty\) or as \(x\to-\infty\). What is the geometrical significance of the last result?

State and prove the Heine-Borel theorem for one variable. Deduce that if \(f(x)\) is continuous in \(a\le x\le b\), then, corresponding to any positive \(\epsilon\), there is an \(\eta\) such that in any interval contained in \(ab\) and of length less than \(\eta\) the oscillation of \(f(x)\) is less than \(\epsilon\).

State and prove Cauchy's Integral test for the convergence of series of positive terms, and deduce the logarithmic scale of convergent and divergent series. Discuss the convergence or divergence of the series whose \(n\)th terms are \[ \frac{n+2\sqrt{n}}{n^2-1}, \quad \frac{(\log n)^3}{n(\log n)^2}, \quad e^{-(\log n)^\alpha}. \]

Prove that if \(D_n\) be any one of the functions \[ 1, n, n\log n, n\log n \log\log n, \dots, \] then a series \(\Sigma a_n\) of positive terms is convergent if \[ \lim\left(D_n \frac{a_n}{a_{n+1}} - D_{n+1}\right)>0. \] Show that if this inequality is true for any \(D_n\) of the sequence, it is true for all subsequent \(D_n\). Discuss the series \[ \Sigma \frac{1 \cdot 3 \dots (2n-1)}{2 \cdot 4 \dots 2n} \frac{1}{\sqrt{n}}. \]

Show that if \(f^{(r)}(x)\) exists at all points of \(a

State and prove Cauchy's theorem on the integral of an analytic function round a closed contour.