The electron mobility $\mu_n$ in a non-degenerate germanium semiconductor at 300 K is $0.38 \mathrm{~m}^2 / \mathrm{Vs}$.
The electron diffusivity $D_n$ at 300 K (in $\mathrm{cm}^2 / \mathrm{s}$, rounded off to the nearest integer) is ____________
(Consider the Boltzmann constant $k_B=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K}$ and the charge of an electron $e=1.6 \times 10^{-19} \mathrm{C}$.)
Consider the matrix $A$ below:
$$ A=\left[\begin{array}{llll} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & \gamma \end{array}\right] $$
For which of the following combinations of $\alpha, \beta$ and $\gamma$, is the rank of $A$ at least three?
(i) $\alpha=0$ and $\beta=\gamma \neq 0$
(ii) $\alpha=\beta=\gamma=0$
(iii) $\beta=\gamma=0$ and $\alpha \neq 0$
(iv) $\alpha=\beta=\gamma \neq 0$
Consider the following series:
(i) $\sum\limits_{n=1}^{\infty} \frac{1}{\sqrt{n}}$
(ii) $ \sum\limits_{n=1}^{\infty} \frac{1}{n(n+1)}$
(iii) $\sum\limits_{n=1}^{\infty} \frac{1}{n!}$