Sequences and Series

Functions: Provide definitions for each of the following:

•  function
•  the domain of a function
•  the codomain of a function
•  the function $f$ is increasing on interval $\left[a,b\right]$
•  the function $f$ is strictly increasing on interval $\left[a,b\right]$
•  the function $f$ is decreasing on interval $\left[a,b\right]$
•  the function $f$ is strictly decreasing on interval $\left[a,b\right]$
•  the function $f$ is constant on interval $\left[a,b\right]$
•  the function $f$ is bounded on interval $\left[a,b\right]$

Provide definitions for each of the following: 

1). function

2). the domain of a function

3). the codomain of a function

4). the function f is increasing on interval [a, b]

5). the function f is strictly increasing on interval [a, b]

6). the function f is decreasing on interval [a, b]

7). the function f is strictly decreasing on interval [a, b]

8). the function f is constant on interval [a, b]

9). the function f is bounded on interval [a, b]

Problem Zero: Read the section and make your own summary of the material. 

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample. 

(a) True or False: Every sequence is a function. 

(b) True or False: The third term of the sequence \(\left \{ k+1 \right \}_{k=1}^{\infty }\) is \(4\). 

(c) True or False: The third term of the sequence \(\left \{ k^2 \right \}_{k=2}^{\infty }\) is \(9\). 

(d) True or False: Every sequence of real numbers is either increasing or decreasing. 

(e) True or False: Every sequence of numbers has a smallest term. 

(f) True or False: Every recursively defined sequence has an infinite number of distinct outputs. 

(g) True or False: Every sequence has an upper bound, a lower bound, or both an upper bound and a lower bound. 

(h) True or False: Every monotonic sequence has an upper bound, a lower bound, or both an upper bound and a lower bound. 

What is a sequence? 

What is a term of a sequence? 

What is meant by the index of a term of a sequence? 

Give a recursive definition for K! for integers k ≥ 0. Be sure you define 0! as part of your answer. 

Give the first five terms of the following recursively defined sequence:

 $a_1=1$, and $a_k=a_{k-1}+2$ for $k\ge 2$.

Also, give a closed formula for the sequence. 

Give the first five terms of the following recursively defined sequence: 

$a_1=2$, and $a_k=a_{k-1}+2$  for $k\ge 2$.

Also, give a closed formula for the sequence.

Give a recursive definition for the sequence $1,2,3,4,....$ of positive integers. (Hint: Let $a_1=1$.)

The Lucas numbers are defined recursively as follows:

$L_1=1,L_2=3,$ and $L_k=L_{k-2}+L_{k-1}$ for $k\ge 3$.

What are $L_3,L_4,L_5,$ and $L_6$?

Define what it means for a sequence $\left\{a_k\right\}$to be eventually strictly increasing.

Define what it means for a sequence $\left\{a_k\right\}$to be eventually decreasing.

Define what it means for a sequence $\left\{a_k\right\}$ to be eventually strictly decreasing.

Define what it means for a sequence $\left\{a_k\right\}$ to be eventually monotonic.

What does it mean for a sequence $\left\{a_k\right\}$ to be bounded above? Bounded below? Bounded?

Explain why a sequence that is bounded above has infinitely many upper bounds.

Give an example of a sequence with neither an upper bound nor a lower bound. 

Explain why every monotonic sequence has an upper bound, a lower bound, or both an upper bound and a lower bound.

Explain why we require the terms of the sequence$a_k$ to be positive when we use the ratio test from Theorem 7.6.

State a variation of the ratio test from Theorem 7.6 that would allow you to use ratios to test a sequence $\left\{a_k\right\}$for monotonicity when each $a_k<0.$

The Fibonacci numbers may be computed with the formula

$f_k=\frac{{\left(1+\sqrt{5}\right)}^k-{\left(1-\sqrt{5}\right)}^k}{2^k\sqrt{5}}$

Use this formula to compute f$f_1, f_2, f_3, f_4, \mathrm{and} f_5.$(Imagine computing $f_{100}$.)

What is the least upper bound property for nonempty subsets of real numbers? Does the least upper bound property hold for subsets of the rational numbers? Does it hold for subsets of the integers?

Make a statement expressing a property analogous to the least upper bound property for nonempty subsets of real numbers that are bounded below.

Let \(\left \{ a_k \right \}\) be the sequence \(a_1=3, a_2=3.1, a_3=3.14, a_4=3.141, ...\) That is, each term \(a_k\) contains the first \(k\) decimal digits of \(\pi\). 

(a) Explain why \(a_k\) is a rational number for each positive integer \(k\). 

(b) Explain why the sequence \(\left \{ a_k \right \}\) is increasing. 

(c) Provide an upper bound for the sequence \(\left \{ a_k \right \}\). 

(d) What is the least upper bound of the sequence \(\left \{ a_k \right \}\) ? 

(e) Use this sequence to explain why the Least Upper Bound Axiom does not apply to the set of rational numbers. 

In Exercises, find a plausible formula for the general term of the given sequence.

$\left\{0,1,0,1,0,1,\dots \right\}$

In Exercises, find a plausible formula for the general term of the given sequence.

$\left\{1,7,13,19,25,\dots \right\}$

In Exercises, find a plausible formula for the general term of the given sequence.

$\left\{\frac{1}{3},\frac{2}{9},\frac{1}{9},\frac{4}{81},\frac{5}{243},\dots \right\}$

In Exercises, find a plausible formula for the general term of the given sequence.

$\left\{5,-\frac{5}{2},\frac{5}{4},-\frac{5}{8},\frac{5}{16},\dots \right\}$

In Exercises, find a plausible formula for the general term of the given sequence.

$\left\{\frac{2}{3},\frac{3}{5},\frac{4}{7},\frac{5}{9},\dots \right\}$

In Exercises, find a plausible formula for the general term of the given sequence.

$\left\{1,\frac{1}{2},\frac{1}{6},\frac{1}{24},\frac{1}{120},\dots \right\}$

In Exercises 31–36 provide the first five terms of the given sequence. Unless specified, assume that the first term has index 1. 

$\left\{\frac{1-(-1{)}^k}{k}\right\}$

In Exercises 31–36 provide the first five terms of the given sequence. Unless specified, assume that the first term has index 1. 

$\left\{\frac{n}{2n+1}\right\}$

In Exercises 31–36 provide the first five terms of the given sequence. Unless specified, assume that the first term has index 1. 

$a_k=\frac{\cos \left(kx\right)}{x^k+k^2}$

In Exercises 31–36 provide the first five terms of the given sequence. Unless specified, assume that the first term has index 1. 

$\left\{\frac{(-1{)}^{k-1}{x}^{2k}}{2k!}\right\}$

In Exercises 31–36 provide the first five terms of the given sequence. Unless specified, assume that the first term has index 1. 

$\left\{\frac{\sqrt{k}}{k}\right\}$

In Exercises 31–36 provide the first five terms of the given sequence. Unless specified, assume that the first term has index $1$. 

$\left\{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots +\frac{1}{2^k}\right\}$

Find the least upper bound of the sequences in Exercises 37–42 

$\left\{2-\frac{1}{k^2}\right\}$

Find the least upper bound of the sequences in Exercises 37–42 

$\left\{\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\dots \right\}$

Find the least upper bound of the sequences in Exercises 37–42 

$\{-k\}$

Find the least upper bound of the sequences in Exercises 37–42 

$\{0,1,0,1,0,1,\dots \}$

Find the least upper bound of the sequences in Exercises 37–42 

$\left\{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots +\frac{1}{2^k}\right\}$

Find the least upper bound of the sequences in Exercises 37–42 

$\{2,2.7,2.71,2.718,2.7182,\dots \}$

In Exercises 43–46 give the first five terms for a geometric sequence $\left\{c{r}^k\right\},k=0$ with the specified values of $c and r.$

$c=3,r=\frac{1}{2}$.

In Exercises 43–46 give the first five terms for a geometric sequence ${\left\{c{r}^k\right\}}_{k=0}^{\infty }$ with the specified values of $c and r.$

$c=-2,r=-\frac{1}{3}$

In Exercises 43–46 give the first five terms for a geometric sequence ${\left\{c{r}^k\right\}}_{k=0}^{\infty }$ with the specified values of $c and r.$

$c=-2, r=-3$.

In Exercises 43–46 give the first five terms for a geometric sequence ${\left\{c{r}^k\right\}}_{k=0}^{\infty }$ with the specified values of $c and r.$

$c=-1,r=-\frac{1}{2}$

Write each of the arithmetic sequences in Exercises 47–50 in the form $\{c+dk{\}}_{k=0}^{\infty }$

$-3,7,17,27,\dots$

Write each of the arithmetic sequences in Exercises 47–50 in the form $\{c+dk{\}}_{k=0}^{\infty }$

$-6,-7.1,-8.2,-9.3,\dots$