Solutions - 0140-06

(a) diverges $$ (b) $1$ $$ (c) diverges $$ (d) ${\displaystyle \frac{3}{2}}$

(e) $0$

(f) diverges:

$$\begin{array}{c}\ln 5^n-\ln n!\\ \\ \gg \gg \ln \left(\frac{5^n}{n!}\right)\stackrel{n\to \infty }{⟶}\ln (0^{+})=-\infty \end{array}$$

(g) $2^{1/3}$

(h) $0$

Use L’Hopital’s Rule:

$$\begin{array}{c}n\sin \frac{\pi }{n}\gg \gg \frac{\sin \frac{\pi }{n}}{1/n}\gg \gg \frac{\sin \frac{\pi }{x}}{1/x}\\ \\ {\underset{d/dx}{⟶}}^{d/dx}\frac{(-\pi /x^2)\cdot \cos \frac{\pi }{x}}{-1/x^2}\gg \gg \pi \cos \frac{\pi }{x}\stackrel{x\to \infty }{⟶}\pi \end{array}$$