Problem 1
Let $x_1 = 1$, $x_2 = 3$, and $\displaystyle{ x_{n+1} = \frac{1}{n + 1} \sum_{i = 1}^n x_i \ \textrm{ for} \ n = 2, 3, … }$
Find $\displaystyle{ \lim_{n \to \infty} x_n }$ and give a proof of your answer.
Problem 2
Given that $a > 0$ and $c > 0$, find a necessary and sufficient condition on $b$ so that $ax^2 + bx + c > 0$ for all $x > 0$.
Problem 3
Express $\sinh 3x$ as a polynomial in $\sinh x$. As an example, the identity $$\cos 2x = 2 \cos ^2 x - 1$$ shows that $\cos 2x$ can be expressed as a polynomial in $\cos x$. Recall that $\sinh$ denotes the hyperbolic sine defined by: $$ \sinh x = \frac{e^x - e^{-x}}{2} $$
Problem 4
Find the quadratic polynomial $p(t) = a_0 + a_1 t + a_2 t^2$ such that:
$$ \int_0^1 t^n p(t) dt = n \quad \textrm{for} \ n = 0, 1, 2. $$
Problem 5
Verify that, for $f(x) = x + 1$,
$$ \lim_{r \to 0^+} \left( \int_0^1 \left( f(x) \right)^r dx \right)^{\frac{1}{r}} = e^{\int_0^1 \ln f(x) dx} $$
Problem 6
Sets $A$ and $B$ are defined by $A = \lbrace 1, 2, …, n \rbrace$ and $B = \lbrace 1, 2, 3 \rbrace$. Determine the number of distinct functions from $A$ onto $B$.
Recall that a function $f : A \rightarrow B$ is “onto” if for each $b \in B$, there exists $a \in A$ such that $f(a) = b$.
Problem 7
A function $f$ from the positive integers to the positive integers has the properties:
- $f(1) = 1$,
- $f(n) = 2$ if $n \ge 100$,
- $f(n) = \displaystyle{ f \left( \frac{n}{2} \right) }$ if $n$ is even and $n < 100$,
- $f(n) = f(n^2 + 7)$ if $n$ is odd and $n > 1$.
(a) Find all positive integers $n$ for which the stated properties require that $f (n) = 1$.
(b) Find all positive integers $n$ for which the stated properties do not determine $f (n)$.
Problem 8
Find all pairs $(M, N)$ of positive integers, $M < N$, such that:
$$ \sum_{j = M}^N \frac{1}{ j (j + 1) } = \frac{1}{10} $$