Question

1. the following sequences of numbers each have a limit. that is, they gradually get closer and closer to a specific number, called the limit. identify the limit of each of the following sequences.

a) 7.1,7.01,7.001,7.0001,... b) 1/2,1/3,1/4,1/5,1/6,... c) 5/1,5/4,5/9,5/16,5/25,...
d) 0.3,0.33,0.333,0.3333,... e) -2.6,-2.66,-2.666,-2.6666,... f) 1.4,1.44,1.444,1.4444,...

1. consider the set {1,1/2,1/4,1/8,1/16,...}

a) describe how the sequence of numbers in this set is constructed.
b) does this set have the density property? explain.
c) does the sequence in this set have a limit? if so, what is the limit? if the sequence does not have a limit, explain why not.

Explanation:

Step1: Analyze sequence a

As the terms progress, the number of zeros after the decimal point increases. The sequence is getting closer and closer to 7.

Step2: Analyze sequence b

The general term of the sequence is $\frac{1}{n}$ where $n$ starts from 2 and increases by 1 each time. As $n$ gets larger, $\frac{1}{n}$ approaches 0.

Step3: Analyze sequence c

The general term is $\frac{5}{n^{2}}$ where $n$ starts from 1 and increases by 1 each time. As $n$ gets larger, $n^{2}$ gets very large and $\frac{5}{n^{2}}$ approaches 0.

Step4: Analyze sequence d

The sequence is $0.\underbrace{33\cdots3}_{n - \text{times}}$. It can be written as $\sum_{i = 1}^{n}\frac{3}{10^{i}}$. As $n$ approaches infinity, it converges to $\frac{1}{3}$.

Step5: Analyze sequence e

The sequence is $- 2.\underbrace{66\cdots6}_{n - \text{times}}$. It is approaching $-\frac{8}{3}$.

Step6: Analyze sequence f

The sequence is $1.\underbrace{44\cdots4}_{n - \text{times}}$. It is approaching $\frac{4}{3}$.

Step7: Analyze set in 11

a

The first - term is 1, and each subsequent term is obtained by dividing the previous term by 2.

b

The density property states that between any two distinct real numbers, there is another number in the set. This set does not have the density property because there are intervals (e.g., between $\frac{1}{2}$ and 1) that do not contain any other elements of the set.

c

The general term of the sequence is $\frac{1}{2^{n - 1}}$ where $n=1,2,\cdots$. As $n$ approaches infinity, $\frac{1}{2^{n - 1}}$ approaches 0.

Answer:

10)
a) 7
b) 0
c) 0
d) $\frac{1}{3}$
e) $-\frac{8}{3}$
f) $\frac{4}{3}$
11)
a) Each term is half of the previous term.
b) No, there are intervals without set - elements.
c) Yes, the limit is 0.