Define a convergent series. State and prove the theorem used in discussing the convergency of such series as \[ \frac{1}{1} - \frac{1}{4} + \frac{1}{7} - \frac{1}{10} + \frac{1}{13} - \dots \] and \[ 2 - \frac{5}{4} + \frac{8}{7} - \frac{11}{10} + \frac{14}{13} - \dots. \] Prove that the second series can be made convergent by bracketing the terms in pairs.
Find from the definition the differential coefficient of \(\sin x\), establishing the limiting value required in the proof. By calculating successive differential coefficients, or otherwise, shew that if \(0 < x < \frac{\pi}{2}\), then \[ 2x + x \cos x - 3 \sin x > 0. \]
Having given that \[ x(1-x)\frac{d^2y}{dx^2} - (3-10x)\frac{dy}{dx} - 24y = 0, \] prove that \[ x(1-x)\frac{d^{n+2}y}{dx^{n+2}} + \{n-3-2(n-5)x\}\frac{d^{n+1}y}{dx^{n+1}} - (n-3)(n-8)\frac{d^ny}{dx^n} = 0. \] Hence find, by Maclaurin's Theorem, that value of \(y\) which is zero when \(x=0\), and is such that its fourth differential coefficient is unity when \(x=0\).
Define the curvature of a plane curve, and deduce the expression \[ \pm \frac{d^2y/dx^2}{\{1+(dy/dx)^2\}^{3/2}}, \] for the curvature at a point on the curve \(y=f(x)\). Prove that the centre of curvature at the point \((x, y)\) on the curve \(ay=x^2\) is the point \((-4x^3/a^2, 3y+a/2)\). Find the coordinates of the points where the locus of centres of curvature cuts the original curve, and shew that at these points the curvature of the locus of centres of curvature is \(\sqrt{6}/27a\).
Evaluate \[ \int x \sin^{-1} x \, dx, \quad \int \frac{3x^2+x-1}{(x^2+1)(x+1)^2} \, dx, \quad \int \frac{cx+f}{(ax^2+2bx+c)^3} \, dx. \] Prove that, if \(a\) and \(b\) are positive, then \[ \int_0^\pi \frac{\sin^2 x \, dx}{a^2+b^2-2ab\cos x} = \frac{\pi}{2a^2} \text{ or } \frac{\pi}{2b^2}, \] according as \(a >\) or \(< b\).
Trace the curve \(r = a(\cos\theta + \cos 2\theta)\), and shew that the curve crosses itself at the points \((a/\sqrt{2}, \pm \frac{1}{4}\pi)\). Prove that the area of that portion of the largest loop that is not common to the other loops is \(\sqrt{2}a^2\).
Explain in what sense the Kelvin scale of temperature is ``absolute.'' How is it possible to test the agreement between the Kelvin absolute scale and the scale of the Hydrogen thermometer?
Explain the principle of the ``Throttling Calorimeter'' for measuring the dryness of steam, and why this can only be used to measure dryness values between about \(\cdot 95\) and 1.0. Dry sat. steam at pressure 100 lbs./sq. in. abs. is throttled at the stop valve of an engine to 70 lbs./sq. in., what is the temp. of the steam as it enters the engine?
A cylinder of compressed carbon dioxide contains 2.1 lbs. of gas at pressure 120 lbs./sq. in. and temperature 15° C. The cylinder may only be subjected to an internal pressure of 350 lbs./sq. in. and the temperature is liable to rise to 30° C. What further weight of CO\(_2\) would it be safe to add to the contents of the cylinder?
Explain carefully what you understand by `reversibility' as applied to a heat engine. Why, and in what respects, are actual engines irreversible? Compare a steam engine and a gas engine from the point of view of reversibility.