Sum the series
Prove the rule for forming the convergents to the continued fraction \[ a_1 + \frac{1}{a_2 +} \frac{1}{a_3 +} \dots, \] and prove that these convergents are fractions in their lowest terms. \par Prove that, if \(p/q\) is the fraction in its lowest terms, which is equal to \[ a_1 + \frac{1}{a_2 +} \frac{1}{a_3 +} \dots + \frac{1}{a_n +} \frac{1}{a_{n-1} +} \dots + \frac{1}{a_2 +} \frac{1}{a_1}, \] then \((q^2+1)/p\) is an integer when \(a_1, a_2, \dots a_n\) are all integers.
Prove that \[ 1-\cos^2 A - \cos^2 B - \cos^2 C + 2\cos A \cos B \cos C = 4 \sin S \sin(S-A) \sin(S-B) \sin(S-C) \] where \[ 2S = A+B+C. \] Given that \(a \sin\alpha = b\sin(\alpha+\beta) = c\sin(\alpha+2\beta)\) express the ratios \[ \cos\alpha:\cos(\alpha+\beta):\cos(\alpha+2\beta) \] in terms of \(a, b, c\).
In solving a triangle in which two sides and the included angle are given, shew how to determine the error introduced into the calculated value of the third side by small errors in the given quantities. \par In a triangle \(ABC\) we have given that approximately \[ a=36 \text{ ft.}, \quad b=50 \text{ ft.}, \quad C=\tan^{-1}\frac{3}{4}; \] find what error in the given value of \(a\) will cause an error in the calculated value of \(c\) equal to that caused by an error of \(5'\) in the measurement of \(C\).
Express \((a+b\sqrt{-1})^{c+d\sqrt{-1}}\) in the form \(A+B\sqrt{-1}\) where all the quantities \(a, b, \dots\) are real. \par Prove that \[ \sum_{r=1}^5 \tan^{-1}\left(\frac{1}{\alpha_r+1}\right) = \tan^{-1}\frac{4}{3}, \] where \(\alpha_1, \dots \alpha_5\) are the fifth roots of unity.
Differentiate \(x^{\log x}\), \((\log x)^x\). \par Find the \(n\)th differential coefficient of \(a^x \sin^n x\).
Prove that, if \(f(a)=0\) and \(\phi(a)=0\) and if \(f'(a)\), \(\phi'(a)\) do not both vanish, \[ \operatorname{Lt}_{x \to a} \frac{f(x)}{\phi(x)} = \frac{f'(a)}{\phi'(a)}. \] Evaluate \[ \operatorname{Lt}_{x \to n} \frac{1}{(n^2-x^2)^2}\left\{ \frac{n^2+x^2}{nx} - 2\sin\frac{n\pi}{2}\sin\frac{x\pi}{2} \right\} \] where \(n\) is an odd integer.
Prove the formulae for the radius of curvature of a curve \[ \rho = \frac{r dr}{dp} = \frac{\left\{r^2+\left(\frac{dr}{d\theta}\right)^2\right\}^{\frac{3}{2}}}{r^2+2\left(\frac{dr}{d\theta}\right)^2 - r\frac{d^2r}{d\theta^2}}. \] Any point on a curve is taken as pole and the tangent at it is the initial line, prove that the approximate equation of the curve in the neighbourhood of the origin is \(r=2\rho\theta + \frac{4}{3}\rho\frac{d\rho}{ds}\theta^2\), where \(\rho\) and \(\frac{d\rho}{ds}\) are the values at the origin.
Prove that two curves intersect at the same angle as their inverses. \par Shew that if a point is such that when it is taken as the origin of inversion two given circles invert into equal circles, the locus of the point is a circle.
Prove that the series \[ 1^2 + 2^2x + 3^2x^2 + 4^2x^3 + \dots \] is convergent when \(x\) lies between \(-1\) and \(+1\). \par Shew that if \(x=0.9\) the sum to infinity is 54,100.