Define the upper and lower limits of a function of an integral variable.
If \(f(n)
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.
Prove that if (1) \(f_n(z)\) is, for every positive integral value of \(n\), analytic in a region \(T\), and (2) \(\Sigma f_n(z)\) is uniformly convergent throughout any closed domain \(D\) interior to \(T\), then the sum \(f(z)\) of the series is analytic in \(T\). Deduce that the sum of a power series represents an analytic function inside its circle of convergence. Prove that the function \(\Sigma n^{-s}\), where \(s=\sigma+it\), is analytic for \(\sigma>1\).
Prove that if \(s_n\) is the sum of the first \(n\) terms of the Fourier series of a continuous and periodic function \(f(x)\), then \[ \frac{s_1+s_2+\dots+s_n}{n} \] tends uniformly to \(f(x)\) when \(n\to\infty\). State any extensions of this theorem with which you are familiar. What are the principal theorems concerning the convergence of Fourier series? How are they related to the theorem just stated?