Desmos by Tim - HackMD

# Desmos by Tim --- ## Week 1 --- **Example 2'.** $\{2n\}$ is not bounded above. [Link to Desmos Example 2' (Desmos Prob 1)](https://www.desmos.com/calculator/noeav5nixt) **Desmos Problem 1** In the provided Desmos link, set the slider for K so that $K=5$. What is the corresponding value for $m$ such that $2m>K$? **Counterdefinition 3.** $\{x_n\}$ is not bounded below if, $\forall K\in {\mathbb R}$, $~\exists m\in {\mathbb N}$ such that $x_m<K$ **Example 3'.** $\{-n\}$ is not bounded below. [Link to Desmos for Example 3' (Desmos Prob 2)](https://www.desmos.com/calculator/5dsr7ln4u7) **Desmos Problem 2** In the provided Desmos link, set the slider for K so that $K=-5.5$. What is the corresponding value for $m$ such that $-m<K$? **Counterdefinition 4.** $\{x_n\}$ is not bounded if, $\forall K\in {\mathbb R}$, $\exists m\in {\mathbb N}$ such that $\lvert x_m \rvert$ $> K$. **Example 4'.** $\{(-2)^n\},$ is not bounded. [Link to Desmos for Example 4' (Desmos Prob 3)](https://www.desmos.com/calculator/0zoubulcd7) **Desmos Problem 3** In the provided Desmos link, m is defined in a way to ensure that $|(-2)^m|>K$. Between what values of K does Desmos suggest $m=6$? ___ ## **Week 2** --- **2. Prove that $\{(-1)^nn\}$ is infinitely large** [Link to Desmos Visual for #2 (Desmos Prob 4)](https://www.desmos.com/calculator/pesvxs91pc) **Desmos Problem 4** Let $\{x_n\}=\{(-1)^nn\}$, as defined in Wk2#2. In the provided Desmos link, set $K=15.5$ What is the corresponding value of $m$ such that $|x_m|>K$? **4. Prove that $$x_n=\begin{cases} n,&\text{if $n$ is even}\\ 0,& \text{if $n$ is odd}\end{cases}$$is not infinitely large, but unbounded** *Part 1 - show that the sequence is unbounded* [Desmos Link to week 2 #4 Part 1 (Desmos Prob 5)](https://www.desmos.com/calculator/ghd8sf6rjw) **Desmos Problem 5** In the Desmos link provided, set $K=12$. What is the corresponding value of $m$ such that $|x_m|>K$? You may need to zoom out. *Part 2 - Show that the sequence is NOT infinitely large* [Desmos Link to week 2 #4 Part 2 (Desmos Prob 6)](https://www.desmos.com/calculator/gfsrecqcap) **Desmos Problem 6** In the link provided, set $n=15$. What value of $n^*$ does Desmos highlight such that we know $0=x_{n^*} < K=1$? --- ## **Week 3** --- **3. Prove that $\sup\{1-\frac{1}{n}\}=1$** [Link to Desmos Week3 #3 (Desmos Prob 7)](https://www.desmos.com/calculator/c61guqnssk) **Desmos Problem 7** In the Desmos link provided, set $\varepsilon = 0.1$. Desmos shows us that $\exists ~ n^*$ such that $1-\varepsilon<x_{n^*} \leq 1$. What is the value of $x_{n^*}$, for the $n^*$ that Desmos highlights? Note - you can click on any point to reveal its coordinates. **5. Prove that $\inf\{(0,1)\}=0$** [Link to Desmos Week3 #5 (Desmos Prob 8)](https://www.desmos.com/calculator/fcrtfa39dt) **Desmos Problem 8** - View the Desmos link for Wk3#5. What is the maximum value of $x^*$, as defined in the Desmos link? **7. Ex. 2.1.3 and 2.1.4 in our text.** ***Exercise 2.1.3*** *Is the sequence $\{\frac{(-1)^n}{2n}\}$ convergent? If so, what is the limit?* [Link to Desmos for Exercise 2.1.3 (Desmos Prob 9)](https://www.desmos.com/calculator/0t4fdmexap) Yes, $\{\frac{(-1)^n}{2n}\}$ is convergent, and the limit is 0. **Desmos Problem 9** - View the Desmos link for Exercise 2.1.3. For $\varepsilon=0.03$, what $n^*$ is suggested such that we know $\forall n \geq n^*,~~|x_n-0|<\varepsilon$? --- ## **Week 4** --- **3. p. 50 Ex. 2.1.10 Show that the sequence $\{x_n\}=\{\frac{n+1}{n}\}$ is monotone, bounded, and use Theorem 2.1.10 to find the limit.** [Desmos Link for Week 4 #3 (Desmos Prob 10)](https://www.desmos.com/calculator/mu7fhwxhoo) **Desmos Problem 10** - Theorem 2.1.10 tells us that if $\{x_n\}$ is monotone decreasing and bounded, then $\lim{x_n}=\inf{x_n}$. For what values of $\varepsilon$ does Desmos recommend $n^* = 4$ such that we know $1 \leq x_n^* < 1 + \varepsilon$? ___ ## **Week 5** ___ [Desmos Problem 11 Link](https://www.desmos.com/calculator/2twzldo7u4) **Desmos Problem 11** Open the Desmos link above. Drag a and b to observe that the limit of the sum $x_n + y_n$ is the sum of the limit $a+b$. Set $\varepsilon$ to 0.5. What are the values of $n_1^*, n_2^*,$ and $n^*$ such that $n^* = \max\{n_1^*, n_2^*\}$ guarantees $\forall n \geq n^*$ that $$|(x_n+y_n)-(a+b)|<\varepsilon$$ ___ ## **Week 6/7** ___ [Desmos Problem 12 Link](https://www.desmos.com/calculator/txutl3okye) **Desmos Problem 12** Open the Desmos link provided, which corresponds to example 2.3.3 from our text. You can drag the black dotted line to cut off the head of the sequence. Click "show sup" to view the sequence of supremums ($a_n$), and click "play lim sup" to get a draggable window $\varepsilon$ from 1. Is it possible to find an $\varepsilon$ that satisfies both of the conditions below? $$|a_4 - 1| > \varepsilon$$ and $$|a_n - 1| < \varepsilon ~~\forall n \geq 5$$ If it is possible, give a value for $\varepsilon$. If it is not possible, why not?