Sequences, Series, and Products

Let $A_{1} \geq A_{2} \geq \cdots \geq A_{k} \geq 0$. Evaluate

Note: See also Problem 5.1.11.

Compute

Let $x_{0} = 1$ and

Prove that $x_{\infty } = \lim_{n \rightarrow \infty } x_{n}$ exists, and find its value.

Define a sequence of real numbers $(x_{n})$ by

Show that $(x_{n})$ converges, and evaluate its limit.

Let $\alpha$ be a number in $(0,1)$. Prove that any sequence $(x_{n})$ of real numbers satisfying the recurrence relation

has a limit, and find an expression for the limit in terms of $\alpha$, $x_{0}$ and $x_{1}$.

Let $k$ be a positive integer. Determine those real numbers $c$ for which every sequence $(x_{n})$ of real numbers satisfying the recurrence relation

has period $k$ (i.e., $x_{n+k} = x_{n}$ for all $n$).

Let $a$ be a positive real number. Define a sequence $(x_{n})$ by

Find a necessary and sufficient condition on $a$ in order that a finite limit $\lim_{n \rightarrow \infty } x_{n}$ should exist.

Let $x_{n}$ be a sequence of real numbers so that

Show that $\lim_{n \rightarrow \infty } x_{n} = x$.

Let $a$ and $x_{0}$ be positive numbers, and define the sequence $(x_{n})_{n=1}^{\infty }$ recursively by

Prove that this sequence converges, and find its limit.

Let $x_{1}$ be a real number, $0 < x_{1} < 1$, and define a sequence by $x_{n+1} = x_{n}-x_{n}^{n+1}$. Show that $\liminf_{n \rightarrow \infty } x_{n} > 0$.

Let $f(x) = \frac{1}{4}+x-x^{2}$. For any real number $x$, define a sequence $(x_{n})$ by $x_{0} = x$ and $x_{n+1} = f(x_{n})$. If the sequence converges, let $x_{\infty }$ denote the limit.

The Fibonacci numbers $f_{1}, f_{2}, \ldots$ are defined recursively by $f_{1} = 1$, $f_{2} = 2$, and $f_{n+1} = f_{n}+f_{n-1}$ for $n \geq 2$. Show that

exists, and evaluate the limit.

Note: See also Problem 7.5.20.

Prove that

For $n$ a positive integer, let $H_{n}$ denote the $n^{\text{th}}$ partial sum of the harmonic series

Let $k > 1$ be an integer. Prove that

where $\log k$ is the natural logarithm of $k$, and $C$ is a constant.

Suppose $x_{1}, x_{2}, x_{3}, \ldots$ is a sequence of nonnegative real numbers satisfying

for all $n \geq 1$. Prove that $\lim_{n \rightarrow \infty } x_{n}$ exists.

Let $(a_{n})$ and $(\varepsilon_{n})$ be sequences of positive numbers. Assume that $\lim_{n \rightarrow \infty } \varepsilon_{n} = 0$ and that there is a number $k$ in $(0,1)$ such that $a_{n+1} \leq ka_{n}+\varepsilon_{n}$ for every $n$. Prove that $\lim_{n \rightarrow \infty } a_{n} = 0$.

Prove or disprove (by giving a counterexample), the following assertion: Every infinite sequence $x_{1}, x_{2}, \ldots$ of real numbers has either a nondecreasing subsequence or a nonincreasing subsequence.

Let $b_{1}, b_{2}, \ldots$ be positive real numbers with

Assume also that $b_{1} < b_{2} < b_{3} < \cdots$. Show that the set of quotients $(b_{m}/b_{n})_{1 \leq n < m}$ is dense in $(1,\infty )$.

Which of the following series converges?

Let $a_{1}, a_{2}, a_{3}, \ldots$ be positive numbers.

For each $(a,b,c) \in \mathbb {R}^{3}$, consider the series

Determine the values of $(a,b,c)$ for which the series

For which real numbers $x$ does the infinite series

converge?

For which values of the real number $a$ does the series

converge?

Let $A$ be the set of positive integers that do not contain the digit 9 in their decimal expansions. Prove that

that is, $A$ defines a convergent subseries of the harmonic series.

Let $a_{1}, a_{2}, \ldots$ be positive numbers such that

Prove that there are positive numbers $c_{1}, c_{2}, \ldots$ such that

Evaluate the limit