mathematics

Real numbers y, p, and n (all greater than 1) satisfy $(\log_{p^{1/n}} y)(\log_{y^{1/n}} p) = 16$, where the logarithms are taken to the bases $p^{1/n}$ and $y^{1/n}$. The value of...

Consider the two series, $S_A$ and $S_B$, where $S_A = \sum_{n=1}^{\infty} \frac{n^2}{2^n}$ $S_B = 1 + \frac{1}{2} + \frac{1}{8} + \frac{1}{16} + \frac{1}{64} + \frac{1}{128} + \fr...

Consider the following series: (i) $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ (ii) $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$ (iii) $\sum_{n=1}^{\infty} \frac{1}{n!}$ Choose the correct...

The value of the line integral $\int_P^Q(z^2 dx + 3y^2 dy + 2xz dz)$ along the straight line joining the points P (1,1,2) and Q (2,3,1) is

Consider the homogeneous ordinary differential equation $x^2 \frac{d^2y}{dx^2} - 3x \frac{dy}{dx} + 3y = 0$, $x > 0$ with $y(x)$ as a general solution. Given that $y(1) = 1$ and $y...