Question $1$: $6$ Marks Determine, using the ratio test, whether the following series converges or diverges. Carefully justify each conclusion. $($a$)\backslash$sum$\_($n$=1)^(\backslash$infty $)(1)/(2^($n$)).($b$)\backslash$sum$\_($n$=1)^(\backslash$infty $)2^($n$)$ Question $2$: $10$ Marks Find the interval of convergence of the series $\backslash$sum$\_($n$=1)^(\backslash$infty $)($x$^($n$))/($n$)$

Question $3$: $13$ Marks

Consider the function defined by

f$($x$)=\{(\backslash$alpha x$^(2)+\backslash$beta x$,$xsinx$,$x$>0)$:$\}$

for what value$($s$)$ of $\backslash$alpha $,\backslash$beta inR is f continuous and differentiable at x$=0?$