Prove that, if \(x\) is less than unity, \[ \frac{1+4x+x^2}{(1-x)^4} = \sum_{n=1}^{\infty} (n^3 x^{n-1}). \] Prove that, if \(n\) is any positive integer greater than 3, \[ n^3 - \binom{n}{2}(n-2)^3 + \binom{n}{3}(n-3)^3 - \dots = 0. \]
Prove that if \(\frac{p_n}{q_n}\) denotes the \(n\)th convergent to the continued fraction \[ a_1 + \frac{1}{a_2+} \frac{1}{a_3+} \dots, \] then \[ p_n q_{n-1} - p_{n-1}q_n = (-1)^n. \] Express \(\sqrt{7}\) as a continued fraction, and find the first convergent which differs from \(\sqrt{7}\) by less than \(\cdot 001\).
Prove that \[ \begin{vmatrix} bc-a^2 & ca-b^2 & ab-c^2 \\ ca-b^2 & ab-c^2 & bc-a^2 \\ ab-c^2 & bc-a^2 & ca-b^2 \end{vmatrix} = (a^3+b^3+c^3-3abc)^2. \]
Find the \(n\)th differential coefficients of \(x^n e^{ax}\) and \(e^{ax}\sin x\), and shew that the \(n\)th differential coefficient of \(\displaystyle\frac{x}{x^2+1}\) is \[ (-1)^n n! \frac{\cos\left((n+1)\cot^{-1}x\right)}{(x^2+1)^{\frac{n+1}{2}}}. \]
Prove that the values of \(x\) which make \(f(x)\) a maximum or a minimum must be such as to satisfy \(f'(x)=0\). Give an example in which a root of \(f'(x)=0\) does not give a maximum or minimum value of \(f(x)\). Through a fixed point \(O\) within an ellipse chords \(POP', QOQ'\) are drawn at right angles to each other. Determine when the product \(OP \cdot OP' \cdot OQ \cdot OQ'\) is a maximum or a minimum.
Shew how to determine the radius of curvature at the origin of a curve given by \(f(x,y)=0\). Find the radius of curvature at the origin of the curve given by \[ (x-y)(x-a)(y-a)^2 = a^2(x+y). \] Find the asymptotes of this curve, and trace the curve.
Integrate \[ \int \frac{dx}{x^4+a^4}, \quad \int \frac{dx}{x^3+a^3}, \quad \int_0^{\frac{\pi}{2}} \sin^6 x dx. \]
Shew how to find the area of a closed curve, whose equation in polar coordinates is given. Find the area of a loop of the curve \(r=a\cos^2 n\theta\), where \(n\) is a positive integer.
Factorize
If \(n\) be an integer and \(P_n\) the product of all the coefficients in the expansion of \((1+x)^n\), prove that \(P_{n+1}/P_n = (n+1)^n/n!\). Prove that the coefficient of \(x^n\) in the expansion of \[ (1+x^{2^n})/(1-x) \text{ is } 2^{n-1}(n^2+4n+2). \] This seems to be a typo in the original paper, as the question is non-sensical. A likely intended question is "Prove that the coefficient of \(x^n\) in the expansion of \((1+x)^{2n}/(1-x)\) is ...". The OCR seems to match the original, however.