Sum the series \[ \sum_{r=1}^n \frac{1}{r(r+1)(r+2)}, \quad \sum_{r=1}^\infty \frac{r}{2^r}, \quad \sum_{r=1}^\infty \frac{1.3.5 \dots (2r-1)}{3.6.9 \dots 3r}. \]
Prove that the inverse of a circle (with respect to a coplanar circle) is a circle or a straight line. Circles \(S\) are drawn to touch two given coplanar intersecting circles \(C_1, C_2\). Show that, if two of the circles \(S\) touch, their point of contact (supposed not on \(C_1\) or \(C_2\)) lies in general on one of two fixed circles.
Describe and prove the funicular polygon method of finding graphically the line of action of the resultant of a set of co-planar non-concurrent forces. Consider, in particular, the case of parallel forces.
Prove one of the following theorems and deduce the other from it.
Prove that, if \(a>0\) and \(ac-b^2>0\), the expression \(ax^2+2bx+c\) is positive for all real values of \(x\). Prove that the expression \[ 10x^2 - 6xy + y^2 - 8x + y + 7 \] is positive for all real values of \(x\) and \(y\).
Prove that the reciprocal of a conic with respect to a focus is a circle. A variable chord of a fixed conic subtends a right angle at a given point \(O\) in the plane of the conic but not on the curve. Show that the chord envelops a conic having a focus at \(O\). In what circumstances is the envelope (i) a parabola, (ii) a circle?
A hollow circular cylinder, of weight \(W'\), is made of uniform thin sheet material and is open at both ends, whose planes are perpendicular to the axis; the cylinder rests with its axis vertical on a smooth horizontal plane. Three equal smooth uniform solid spheres, each of weight \(W\), are placed inside the cylinder, and their centres are \(A, B, C\). The lowest sphere \(A\) lies on the table; the sphere \(B\) lies above it, with \(AB\) inclined at an angle \(\alpha\) (\(< \pi/3\)) to the vertical. The third sphere is uppermost, and \(CA\) is vertical. If the reactions of the spheres on the cylinder do not cause it to overturn, show that \(2W \sin\alpha < W'(1 + \sin\alpha)\).
If three conics have double contact in pairs, prove that the extremities of each chord of contact form a harmonic range with the points where this chord meets the other two chords of contact. Prove, conversely, that if pairs of points \((A, A')\), \((B, B')\), \((C, C')\) are taken in a plane so that \(A, A'\) are harmonically conjugate with respect to the points where \(AA'\) meets \(BB'\) and \(CC'\), and similarly for the other pairs, then a set of three conics can in general be found having double contact in pairs at the ends of the segments \(AA', BB', CC'\).
Show that the equation \[ \sin x = \tanh x \] has infinitely many real roots and that, if \(n\) is a large positive integer, approximate values of a pair of roots are \[ x = (2n + \tfrac{1}{2}) \pi \pm 2e^{-(2n+\frac{1}{2})\pi}. \]
Prove Pascal's theorem that, if a hexagon is inscribed in a conic, the meets of pairs of opposite sides are collinear. Two coplanar triangles \(ABC\) and \(XYZ\) are such that \(ABC\) is in perspective with \(YZX\) and also with \(ZXY\) (the vertices being associated in the order written). By considering the two axes of perspective as a conic, or otherwise, prove that \(ABC\) is also in perspective with \(XYZ\).