A leaf of a book is of width \(a\) and height \(b\), where \(3a \leq 2\sqrt{2}b\); the lower corner of the leaf is folded over so that the corner just reaches the inner edge of the page. Find the minimum length of the resulting crease. Explain why the condition \(3a \leq 2\sqrt{2}b\) is relevant.
Prove that, if \(|x| < 1\), then \[x - \frac{1}{2}\left(\frac{2x}{1+x^2}\right) = \frac{1}{2.4}\left(\frac{2x}{1+x^2}\right)^3 + \frac{1.3}{2.4.6}\left(\frac{2x}{1+x^2}\right)^5 + \frac{1.3.5}{2.4.6.8}\left(\frac{2x}{1+x^2}\right)^7 + \cdots\] What function is represented by the infinite series when \(|x| > 1\)?
Solution: Consider \begin{align*} \frac{1 - \sqrt{1-y^2}}{y} &= \frac{1 - \sum_{k=0}^{\infty} \binom{-1/2}{k}(-y^2)^k}{y} \\ &= \frac{1}{y} \left (-\sum_{k=1}^{\infty} \binom{1/2}{k}(-y^2)^k \right) \\ &= -\sum_{k=1}^\infty \frac{\left(\frac12 \right)\left(-\frac12 \right)\left(-\frac32 \right)\cdots \left(-\frac{2k-3}2 \right)}{k!} (-1)^ky^{2k-1} \\ &= \sum_{k=1}^\infty \frac{\left (\frac12 \right)\left(\frac12 \right)\left(\frac32 \right)\cdots \left(\frac{2k-3}2 \right)}{k!} y^{2k-1} \\ &= \sum_{k=1}^\infty \frac{1 \cdot 3 \cdots (2k-3)}{2 \cdot 4 \cdots (2k)} y^{2k-1} \\ \end{align*} But since \(\frac{1-\sqrt{1-y^2}}{2}\) is the inverse for \(\frac{2x}{1+x^2}\) for \(|x| < 1\) this computes the inverse. When \(|x| > 1\) it will compute \begin{align*} && \frac{1-\sqrt{1- \left (\frac{2x}{1+x^2} \right)^2}}{\frac{2x}{1+x^2}} &= \frac{1 - \sqrt{\frac{(1-x^2)^2}{(1+x^2)^2}}}{\frac{2x}{1+x^2}} \\ &&&= \frac{1 - \frac{x^2-1}{1+x^2}}{\frac{2x}{1+x^2}} \\ &&&= \frac{1}{x} \end{align*} and in particular the series is equal to \(\frac{1}{x} - \frac{1}{2}\left(\frac{2x}{1+x^2}\right)\)
By means of the calculus or otherwise, prove that if \(p > q > 0\) and \(x > 0\), then \[q(x^p - 1) > p(x^q - 1).\] Hence or otherwise prove that, under the same conditions, \[\frac{1}{p}\left(\frac{x^p}{(p+1)^p} - 1\right) > \frac{1}{q}\left(\frac{x^q}{(q+1)^q} - 1\right)\] for every positive integer \(n\).
Prove that, if \(x > 0\) and \(N\) is a positive integer, then \[\frac{1}{2^x} + \frac{1}{3^x} + \cdots + \frac{1}{(N+1)^x} < \int_1^{N+1} \frac{dx}{x^x} < 1 + \frac{1}{2^x} + \cdots + \frac{1}{N^x}.\] Deduce, or prove otherwise, that \(\sum_{n=1}^{\infty} n^{-x}\) is convergent when \(x > 1\) and divergent when \(x < 1\). Find the set of values of the real number \(\beta\) for which the infinite series \[\sum_{n=1}^{\infty} \frac{n^{\beta}}{n^{2\beta} - n^{\beta} + 1}\] is convergent, and the set of values of \(\beta\) for which it is divergent.
(i) A groove of semicircular section, of radius \(b\), is cut round a right circular cylinder of radius \(a\), where \(a > b\); find the surface area of the groove. (ii) Suppose that the region \(R\) of area \(A\) in the first quadrant of the \((x, y)\)-plane generates a solid of revolution of volume \(U\) when it is revolved about the \(x\)-axis, and a solid of revolution of volume \(V\) when it is revolved about the \(y\)-axis. Find the volume generated by \(R\) when it is revolved about the straight line whose equation is \[x\cos\alpha + y\sin\alpha = p,\] assuming that this line does not meet \(R\). Explain why the sign of the expression obtained for the volume appears to be negative for certain positions of the line, and describe the positions for which this happens.