Math 126B

Worksheet #1

Spring 2003

Names:

(1) Before we begin this worksheet, it's time to get to know the members of your group. Which

member knows the most or all of this week's seven Iinalists on American Idol?

(2) Let's take a moment to get familiar with sequences. Develop the formula for each of the three

sequences:

(a) {2, 4, 8, 16, $\mathrm{\text{...}}$} ${\mathit{a}}_{\mathit{n}}\mathrm{=}$

(b) $\mathrm{\{}\mathrm{2}\mathrm{,}$ $\mathrm{-}\mathrm{4}\mathrm{,}\mathrm{}\mathrm{8}\mathrm{,}$ $\mathrm{-}\mathrm{1}\mathrm{6}\mathrm{,}\mathrm{}\mathrm{\text{...}}\mathrm{\}}$ ${\mathit{b}}_{\mathit{n}}\mathrm{=}$

(c) $\mathrm{\{}\mathrm{1}\mathrm{,}\mathrm{}\mathrm{7}\mathrm{,}$ $\mathrm{-}\mathrm{3}\mathrm{,}\mathrm{}\mathrm{1}\mathrm{1}\mathrm{,}$ $\mathrm{-}\mathrm{7}\mathrm{,}\mathrm{}\mathrm{1}\mathrm{5}\mathrm{,}$ $\mathrm{-}\mathrm{1}\mathrm{1}\mathrm{\text{...}}\mathrm{\}}$ ${\mathit{c}}_{\mathit{n}}\mathrm{=}$

(Note: If you're being challenged, come back to (c) when you Iinish the other problems.)

A popular method for mathematicians is called induction. A superficial perspective is that

induction proves something works when (1) the initial case works and (2) the follow up cases

work. We will want to use induction in order to show that the following expression has a limit:

$\sqrt{\mathrm{1}\mathrm{+}\sqrt{\mathrm{1}\mathrm{+}\sqrt{\mathrm{1}\mathrm{+}\sqrt{\mathrm{1}\mathrm{+}}}}}\mathrm{\text{...}}$.

Let's use the Monotonic Sequence Theorem to prove this expression has a limit: Every bounded,

monotonic sequence is convergent. (Recall, convergent implies that the expression has a limit.) To

use the theorem, we have to show that the expression is bounded and is monotonic.

(3) First, let's get familiar with the expression. We can view the expression has the following

sequence: ${\mathit{b}}_{\mathit{n}\mathrm{+}\mathrm{1}}\mathrm{=}\sqrt{\mathrm{1}\mathrm{+}{\mathit{b}}_{\mathit{n}}}$.

(a) Compute the values for ${\mathit{b}}_{\mathrm{1}}$, ${\mathit{b}}_{\mathrm{2}}$, ${\mathit{b}}_{\mathrm{3}}$, ${\mathit{b}}_{\mathrm{4}}$, and ${\mathit{b}}_{\mathrm{5}}$, where ${\mathit{b}}_{\mathrm{0}}\mathrm{=}\mathrm{0}$.

${\mathit{b}}_{\mathrm{1}}\mathrm{=}$ ${\mathit{b}}_{\mathrm{2}}\mathrm{=}$ ${\mathit{b}}_{\mathrm{3}}\mathrm{=}$

${\mathit{b}}_{\mathrm{4}}\mathrm{=}$ ${\mathit{b}}_{\mathrm{5}}\mathrm{=}$

(b) Looking at just the Iirst Iive terms, determine whether your intuition indicates whether

the sequence is bounded and monotonic. Explain.

Math 126B

Worksheet #1

Spring 2003

Bounded?

Monotonic?

(4) Lets' prove that ${\mathit{b}}_{\mathit{n}}$ is actually bounded. To do that, pick a number that you think bounds

the sequence at the top.

Now, is ${\mathit{b}}_{\mathrm{0}}$ less than your number? (I hope that you picked a number bigger than 0 and bigger

than ${\mathit{b}}_{\mathrm{5}}\mathrm{.}$) That satisfies the Iirst step of induction; the initial value is less than your number.

Next, assume that ${\mathit{b}}_{\mathit{i}}$ is less than your number. You now want to do some algebra to show

that ${\mathit{b}}_{\mathit{i}\mathrm{+}\mathrm{1}}$ is also less than your number. In notation form, assume ${\mathit{b}}_{\mathit{i}}\mathrm{<}\mathit{M}$. Show ${\mathit{b}}_{\mathit{i}\mathrm{+}\mathrm{1}}\mathrm{<}\mathit{M}$.

(Hint: work both sides of the inequality.)

Math 126B

Worksheet #1

Spring 2003

(5) Now that $\mathrm{w}\mathrm{e}\mathrm{\text{'}}\mathrm{v}\mathrm{e}$ proved ${\mathit{b}}_{\mathit{n}}$ is bounded, let's show that it's monotonically increasing. $\mathrm{W}\mathrm{e}\mathrm{\text{'}}\mathrm{1}\mathrm{1}$

following a similar pattern.

Is ${\mathit{b}}_{\mathrm{0}}\underset{\mathrm{‾}}{\mathrm{<}}{\mathit{b}}_{\mathrm{1}}$ ? So, $\mathrm{w}\mathrm{e}\mathrm{\text{'}}\mathrm{v}\mathrm{e}$ got the initial case increasing.

Next, assume that ${\mathit{b}}_{\mathit{i}}\underset{\mathrm{‾}}{\mathrm{<}}{\mathit{b}}_{\mathit{i}\mathrm{+}\mathrm{1}}$. Show that ${\mathit{b}}_{\mathit{i}\mathrm{+}\mathrm{1}}\underset{\mathrm{‾}}{\mathrm{<}}{\mathit{b}}_{\mathit{i}\mathrm{+}\mathrm{2}}$. (Once again, work both sides of the

inequalities to make the thing you want.)

(6) Since you've shown that ${\mathit{b}}_{\mathit{n}}$ is bounded and monotonically increasing, we know that

${\mathit{b}}_{\mathit{n}}$ converges and has a limit. It's time to determine the limit. Begin with

${\lim }_{\mathit{x}\mathrm{\to }\mathrm{\infty }}{\mathit{b}}_{\mathit{i}\mathrm{+}\mathrm{1}}\mathrm{=}{\lim }_{\mathit{x}\mathrm{\to }\mathrm{\infty }}\sqrt{\mathrm{1}\mathrm{+}{\mathit{b}}_{\mathit{i}}}$. Use the rules for limits and also use ${\lim }_{\mathit{x}\mathrm{\to }\mathrm{\infty }}{\mathit{b}}_{\mathit{i}}\mathrm{=}\mathit{L}$. Then, solve for

$\mathit{L}$.