# Graduate preliminary examination, Spring 2007 3 hours, 8 problems counted equally

Write the following statement in symbolic form, and then give (in symbolic form and in English) its contrapositive and its negation:

"If all birds can swim or some fish can fly, then no whales can walk"

Using mathematical induction, show that for each positive integer ${ \mathfrak { n } } ,$

$$
1 ^ { 2 } - 2 ^ { 2 } + 3 ^ { 2 } - \cdots + ( - 1 ) ^ { n - 1 } n ^ { 2 } = ( - 1 ) ^ { n - 1 } n ( n + 1 ) / 2
$$

(Hint: in the induction step, consider separately the cases where n is even or odd.)

Give the $\varepsilon - \delta$ definition of the one-sided limit $\operatorname* { l i m } _ { x \to a ^ { + } } f ( x ) = L$

Using the definition and basic properties of ln(x), show that $\operatorname* { l i m } _ { x  0 ^ { + } } 1 / \ln ( x ) = 0$

Find the Taylor expansion of $f ( x ) = \ln ( 2 + x )$ about the origin.

Find the radius of convergence R of that series;

Use Taylor's theorem to show that the series converges to(xo $[ 0 , R / 2 ]$

5) Let γ be the closed path which goes counterclockwise around the circle $C ( 0 , 2 ) = \{ ( x , y ) \in R ^ { 2 } : x ^ { 2 } + y ^ { 2 } = 2 ^ { 2 } \}$ from the point (2,0) to the point (0,2), then goes to the origin along the y-axis, and then back to (2,0) along the x-axis.

Compute $\int _ { \gamma } 2 x d y + y d x$ in one of two ways:

Directly, using the definition of a line integral; or

By using Green's theorem or the general Stokes' theorem.

Give an example (proof not required) of each of the following:

aA function $f : \mathcal { R } \to \mathcal { R }$ which satisfies $f ( x ) = 0$ for all $x \leq 0$ , but which is nonzero for $x > 0$ and has derivatives of all orders for all x.

An infinite-dimensional vector space over the field $\mathbb { Z } / 2 \mathbb { Z }$

A matrix in $M _ { \mathfrak { z } } ( \mathbb { C } )$ which is not diagonalizable.

A power series $\sum _ { n = 0 } ^ { \infty } a _ { n } z ^ { n } \in { \mathfrak { C } } [ [ z ] ]$ with radius of convergence0.

Let $A = \left[ \begin{array} { l l l } { 1 } & { - 1 } & { 2 } \\ { - 1 } & { 2 } & { - 5 } \\ { 2 } & { - 1 } & { 1 } \end{array} \right] \in M _ { 3 } ( k )$ , and let $T _ { A } : \mathcal { R } ^ { 3 }  \mathcal { R } ^ { 3 }$ be the linear transformation

$\vec { T _ { A } ( x ) } = A \vec { x }$ .Find $\dim _ { \mathit { a R } } ( K e r ( A ) )$ , and use this to determine $\dim _ { \mathit { n } _ { R } } ( \operatorname { I m } ( A ) )$

State the Spectral Theorem (overR).

Let $A = \left[ { \begin{array} { l l } { 2 } & { - 2 } \\ { - 2 } & { 5 } \end{array} } \right]$ .Find an orthogonal matrix P for which $P ^ { - 1 } A P$ is diagonal.