• Work 5 out of 6 problems.
• Each problem is worth 20 points.
• Write on one side of the paper only and hand your work in order.
• Do not interpret a problem in such a way that it becomes trivial.

(1) Determine whether the following sequences converge and carefully justify your claims: (i) $x _ { n } = { \frac { 2 n \cdot n ! } { n ^ { n } } } ;$ (ii) $y _ { n } = \sum _ { k = 1 } ^ { n } { \frac { \cos ( k ! ) } { k ( k + 1 ) } } .$

(2) Find the domain of convergence and the sum of the series

$$
\sum _ { n \geq 0 } ( - 1 ) ^ { n } { \frac { x ^ { 2 n + 1 } } { 2 n + 1 } } .
$$

Show how one may use the sum of the series to provide an approximation for π up to three decimals.
Be sure to provide all technical details.

(3) Define $f , \alpha \in B ( [ - 2 , 2 ] )$ by

$$
f ( x ) : = \left\{ \begin{array} { l l } { - 1 , } & { x \in [ - 2 , 0 ) ; } \\ { 3 , } & { x \in [ 0 , 2 ] } \end{array} \right. \quad \mathrm { ~ a n d ~ } \quad \alpha ( x ) : = \left\{ \begin{array} { l l } { - 2 , } & { x \in [ - 2 , 0 ] ; } \\ { 1 , } & { x \in ( 0 , 2 ] . } \end{array} \right.
$$

Determine whether f is Riemann-Stieltjes integrable with respect to α over $[ - 1 , 1 ]$ . If it is, evaluate $\int _ { - 1 } ^ { 1 } f ( x ) \mathrm { d } \alpha ( x ) .$

(4) (a) Given a set S, show that the function $\rho _ { \infty } : B ( S ) \times B ( S ) $ R defined by

$$
\rho _ { \infty } ( f , g ) : = \operatorname { l u b } \left( \{ | f ( x ) - g ( x ) | : x \in S \} \right)
$$

is a metric on $B ( S )$

(b) Let $M > 0$ be given.
Set

$S : = \{ f \in \mathcal { C } _ { \mathrm { b } } ( [ 0 , 1 ] ) : f ( 0 ) = 0 \}$ , f is differentiable on (0, 1), and $| f ^ { \prime } ( x ) | \leq M$ for each $x \in ( 0 , 1 ) \big \}$

Determine whether the set S is compact in $( \mathcal { C } _ { \mathrm { b } } ( [ 0 , 1 ] ) , \rho _ { \infty } )$

(5) (a) If $\{ f _ { n } \} _ { n \geq 1 }$ is an equicontinuous sequence of functions on a compact interval and $f _ { n }  f$ pointwise, prove that the convergence is uniform.

(b) Let $\alpha , M > 0$ be given, and suppose that $\{ f _ { n } \} _ { n \geq 1 }$ satisfies $| f _ { n } ( x ) - f _ { n } ( y ) | \leq M | x - y | ^ { \alpha }$ 2 for all $n \geq 1$ and all $x , y$ in an interval $[ a , b ]$ . Show that $\{ f _ { n } \} _ { n \geq 1 }$ is equicontinuous.

(6) Let $f : \mathbb { R } \to \mathbb { R }$ be a continuous function, and for all $n \geq 1$ , put $\textstyle f _ { n } ( x ) = f ( x + { \frac { 1 } { n } } )$

(a) Show that $f _ { n }$ converges uniformly, as $n \to \infty$ , over any closed interval $[ a , \dot { b } ]$

(b) Give an example of a continuous function f for which the convergence is not uniform on R.