Logarithms, sequences, and limits

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.title[
# Logarithms, sequences, and limits
]
.author[
### <a href="https://jmclip.github.io/MACSS_math_camp/">MACS 33000</a> <br /> University of Chicago
]

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# Learning objectives

* Define logarithmic and exponential functions
* Practice simplifying power, logarithmic, and exponential functions
* Define sequences
* Distinguish convergence and divergence
* Define limits
* Define continuity
* Calculate limits of sequences and functions

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# Logarithms and exponential functions

* Important component to many mathematical and statistical methods in social science
* Exponents
* Logarithms

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# Functions with exponents

`$$f(x) = x \times x = x^2$$`

`$$f(x) = x \times x \times x = x^3$$`

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# Common rules of exponents

* `$x^0 = 1$`
* `$x^1 = x$`
* `$\left ( \frac{x}{y} \right )^a = \left ( \frac{x^a}{y^a}\right ) = x^a y^{-a}$`
* `$(x^a)^b = x^{ab}$`
* `$(xy)^a = x^a y^a$`
* `$x^a \times x^b = x^{a+b}$`

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# Logarithms

* Class of functions
* `$\log_{b}(x) = a \Rightarrow b^a = x$`
* What number `$a$` solves `$b^a = x$`

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# Base 10

`$$\log_{10}(100) = 2 \Rightarrow 10^2 = 100$$`

`$$\log_{10}(0.1) = -1 \Rightarrow 10^{-1} = 0.1$$`

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# Base 2

`$$\log_{2}(8) = 3 \Rightarrow 2^3 = 8$$`

`$$\log_{2}(1) = 0 \Rightarrow 2^0 = 1$$`

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# Base `$e$`

* Natural logarithm

`$$\log_{e}(e) = 1 \Rightarrow e^1 = e$$`

* Natural logarithms are incredibly useful in math
* Often `$\log()$` is assumed to be a natural log (good to assume for homework, too!)
* Also seen as `$\ln()$`

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# Rules of logarithms

* `$\log_b(1) = 0$`
* `$\log(x \times y) = \log(x) + \log(y)$`
* `$\log(\frac{x}{y}) = \log(x) - \log(y)$`
* `$\log(x^y) = y \log(x)$`

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# Sequence

* A sequence is a function whose domain is the set of positive integers
* We'll write a sequence as,

`$$\left\{u_{n} \right\}_{n=1}^{\infty} = (u_{1} , u_{2}, \ldots, u_{N}, \ldots )$$`

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# Sequence

`$$\left\{\frac{1}{n} \right\} = (1, 1/2, 1/3, 1/4, \ldots, 1/N, \ldots, )$$`

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# Sequence

`$$\left\{\frac{1}{n^2} \right\} = (1, 1/4, 1/9, 1/16, \ldots, 1/N^2, \ldots, )   \\$$`

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# Sequence

`$$\left\{\frac{1 + (-1)^n}{2} \right\} = (0, 1, 0, 1, \ldots, 0,1,0,1 \ldots, ) \\$$`

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# Arithmetic progression

* A sequence `$\{ u_n \}$` with the property that the difference between each pair of successive terms is the same
    * `$u_{n+1} - u_n$` is the same for all `$n$`
* The arithmetic progression with first term `$a$` and common difference `$d$` is

`$$a, a + d, a + 2d, a +3d, \ldots$$`
  
* The `$n$`th term is given by

`$$u_n = a + (n-1)d$$`

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# Arithmetic progression

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# Geometric progression

* A sequence `$\{ u_n \}$` in which each term is obtained from the preceding one by multiplication by the same number:
    * The ratio `$\frac{u_{n+1}}{u_n}$` is the same for all `$n$`
* The geometric progression with first term `$a$` and common ratio `$x$` is

`$$a, ax, ax^2, ax^3, \ldots$$`
  
* The `$n$`th term is given by

`$$u_n = ax^{n-1}$$`

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# Geometric progression

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# Convergence

`$$\left\{\frac{(-1)^{n} }{n} \right \} = (-1, \frac{1}{2}, \frac{-1}{3}, \frac{1}{4}, \frac{-1}{5}, \frac{1}{6}, \frac{-1}{7}, \frac{1}{8}, \ldots )$$`

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# Convergence

A sequence `$\left\{u_{n} \right\}_{n=1}^{\infty}$` converges to a real number `$A$` if for each `$\epsilon >0$` there is a positive integer `$N$` such that for all `$n \geq N$` we have `$|u_{n} - A| < \epsilon$`

* If a sequence converges, it converges to **one** number `$A$`
* `$\epsilon>0$` is some **arbitrary** real-valued number
* `$N$` will depend upon `$\epsilon$`
* Implies the sequence never gets further than `$\epsilon$` away from `$A$`

## Divergence and boundedness

* If a sequence `$\left\{u_{n} \right\}$` converges we'll call it **convergent**
* If it doesn't we'll call it **divergent**
* If there is some number `$M$` such that, for all `$n$` `$|u_{n}|<M$`, then we'll call it **bounded**

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# An unbounded sequence

`$$\left\{ n \right \} = (1, 2, 3, 4, \ldots, N, \ldots )$$`

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# A bounded sequence that doesn't converge

`$$\left\{\frac{1 + (-1)^n}{2} \right\} = (0, 1, 0, 1, \ldots, 0,1,0,1 \ldots, )$$`

* All convergent sequences are bounded
* If a sequence is **constant**, `$\left\{C \right \}$` it converges to `$C$`

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# Algebra of sequences

Suppose `$\left\{a_{n} \right \}$` converges to `$A$` and `$\left\{b_{n} \right\}$` converges to `$B$`. Then,

* `$\left\{a_{n} + b_{n} \right\}$` converges to `$A + B$`
* `$\left\{a_{n} b_{n} \right\}$` converges to `$A \times B$`
* Suppose `$b_{n} \neq 0 \forall n$` and `$B \neq 0$`. Then `$\left\{\frac{a_{n}}{b_{n}} \right\}$` converges to `$\frac{A}{B}$`

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# Algebra of sequences

* Consider the sequence `$\left\{\frac{1}{n} \right\}$` - what does it converge to?
* Consider the sequence `$\left\{\frac{1}{2n} \right \}$` - what does it converge to?

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# Challenge questions

* What does `$\left\{3 + \frac{1}{n}\right\}$` converge to?
* What about `$\left\{ (3 + \frac{1}{n} ) (100  + \frac{1}{n^4} ) \right\}$`?
* Finally, `$\left\{ \frac{ 300 + \frac{1}{n} }{100  + \frac{1}{n^4}} \right\}$`?

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# Limits

* Sequences `$\leadsto$` limits of functions
* Calculus/Real Analysis: study of functions on the **real line**
* Limit of a function: how does a function behave as it gets close to a particular point?

## Uses of limits

* Derivatives
* Asymptotics 
* Game Theory

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# Limit of a function

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# Limits

We need to talk about our functions and how they behave. We're going to be interested in talking about what they do, but we need to make sure they are 'predictable' in a way that makes sense for our analysis.

We've seen how things do/not converge, and so we want to have a way to talk about this in the land of functions.

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## Limit: Visual
We can think about our definition as speaking to what a function is doing around any of its points in the domain. For a limit, if we make some bounds around our variable of interest (e.g. in the domain), and similarly 'arbitrarily small' bounds around our corresponding y, then the graph/function/line for y will lie within the bounds of our range of our variable.

--
<img src="https://math.libretexts.org/@api/deki/files/9980/CNX_Calc_Figure_02_05_001.jpeg?revision=1" width="100%" style="display: block; margin: auto;" />

source: [math 210](https://math.libretexts.org/Courses/Monroe_Community_College/MTH_210_Calculus_I_%28Professor_Dean%29/Chapter_2_Limits/2.7%3A_The_Precise_Definition_of_a_Limit)

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# Limit of a function: Formal (DO NOT NEED TO KNOW THIS!)

* Suppose `$f: \Re \rightarrow \Re$`

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*Pieces of a limit*
`$$|f(x) - L| < \epsilon \,$$` there's an epsilon larger than the distance between f(x) and the limit

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$$\forall \, x \backepsilon $$ for all x, such that,

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`$$0 < |x - x_0 | < \delta$$` the difference between x and `$x_0$` bounded between 0 and some other delta 
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* We say that `$f$` has a limit `$L$` at `$x_{0}$` if, for `$\epsilon>0$`, there is a `$\delta>0$` such that

`$$|f(x) - L| < \epsilon \, \forall \,x \backepsilon 0 < |x - x_0 | < \delta$$`

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# Limits, cont'd

* Limits are about the behavior of functions at **points**. Here `$x_{0}$`
* As with sequences, we let `$\epsilon$` define an **error rate**
* `$\delta$` defines an area around `$x_{0}$` where `$f(x)$` is going to be within our error rate

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# Examples of limits

* The function `$f(x) = x + 1$` has a limit of `$1$` at `$x_{0} = 0$`

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# Proof

* **Without loss of generalization** (WLOG) choose `$\epsilon >0$`
* Show that there is `$\delta_{\epsilon}$` such that

`$$|f(x) - L| < \epsilon \, \text{for all} \, x \, \text{such that} \, 0 < |x - x_0 | < \delta$$`

`$$\begin{aligned}|(x + 1) - 1| < \epsilon \, \text{for all} \, x \, &\text{such that} \, 0 < |x - 0 | < \delta \\|x| < \epsilon \, \text{for all} \, x \, &\text{such that} \, 0 < |x | < \delta \\\end{aligned}$$`

* But if `$\delta_{\epsilon}  = \epsilon$` then this holds

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# `$f(x_{0} ) \neq L$`

* A function can have a limit of `$L$` at `$x_{0}$` even if `$f(x_{0} ) \neq L$`

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* The function `$f(x) = \frac{x^2 - 1}{x - 1}$` has a limit of `$2$` at `$x_{0} = 1$`

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# `$f(x_{0} ) \neq L$`

* For all `$x \neq 1$`,

`$$\begin{aligned}\frac{x^2 - 1}{x - 1} & = \frac{(x + 1)(x - 1) }{x - 1} \\													& = x + 1 \end{aligned}$$`
  
* Choose `$\epsilon >0$` and set `$x_{0}=1$`.  Then, we're looking for `$\delta_{\epsilon}$` such that

`$$\begin{aligned}|(x + 1) -2 | < \epsilon \, \text{for all} \, x \, &\text{such that} \, 0 < |x - 1 | < \delta \\|x - 1 | < \epsilon \, \text{for all} \, x \, &\text{such that} \, 0 < |x - 1 | < \delta \\\end{aligned}$$`

* If `$\delta_{\epsilon} = \epsilon$`, then this is satisfied

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# Not all functions have limits

Consider `$f:(0,1) \rightarrow \Re$`, `$f(x) = \frac{1}{x}$`.  `$f(x)$` does not have a limit at `$x_{0}=0$`

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# Not all functions have limits

* Choose `$\epsilon>0$`. We need to show that there **does not** exist

`$$\begin{aligned}|\frac{1}{x} - L| < \epsilon \, \text{for all} \, x \, &\text{such that} \, 0 < |x - 0 | < \delta \\|\frac{1}{x} - L| < \epsilon \, \text{for all} \, x \, &\text{such that} \, 0 < |x| < \delta \\\end{aligned}$$`

* But, there is a problem

`$$\begin{aligned}\frac{1}{x} - L & < \epsilon \\\frac{1}{x} & < \epsilon + L \\x & > \frac{1}{L + \epsilon}  \end{aligned}$$`
  
* This implies that there **can't** be a `$\delta$`

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# When there are no limits!
<img src="https://media.giphy.com/media/7JvlHfd7C2GDr7zfZF/giphy.gif" width="80%" style="display: block; margin: auto;" />
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When there is no limit, this has a lot of implications for us w/r/t (with respect to) how we can expect the function(s) to behave and what we can do with the function(s) mathematically.
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# Intuitive definition of a limit

`$$\lim_{x \rightarrow x_{0}} f(x) = L$$`

## Right-hand limits

`$$\lim_{x \rightarrow x_{0}^{+} } f(x) = L$$`

## Left-hand limits

`$$\lim_{x \rightarrow x_{0}^{-} } f(x) = L$$`

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# Algebra of limits

Suppose `$f:\Re \rightarrow \Re$` and `$g: \Re \rightarrow \Re$` with limits `$A$` and `$B$` at `$x_{0}$`. Then,

$$
`\begin{aligned}
\text{i.) } \lim_{x \rightarrow x_{0} } (f(x) + g(x) ) & = \lim_{x \rightarrow x_{0}} f(x) + \lim_{x \rightarrow x_{0}} g(x)  = A + B \\
\text{ii.) }\lim_{x \rightarrow x_{0} } f(x) g(x) & = \lim_{x \rightarrow x_{0}} f(x) \lim_{x\rightarrow x_{0}} g(x)  = A B 
\end{aligned}`
$$

Suppose `$g(x) \neq 0$` for all `$x \in \Re$` and `$B \neq 0$` then `$\frac{f(x)}{g(x)}$` has a limit at `$x_{0}$` and

`$$\lim_{x \rightarrow x_{0}} \frac{f(x)}{g(x)} =  \frac{\lim_{x\rightarrow x_{0} } f(x) }{\lim_{x \rightarrow x_{0} } g(x) } = \frac{A}{B}$$`

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# Continuity

* Limit exists at 1 
* But hole in function 
* Fails the **pencil** test, **discontinuous** at 1

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# Defining continuity

* Suppose `$f:\Re \rightarrow \Re$` and consider `$x_{0} \in \Re$`
* `$f$` is continuous at `$x_{0}$` if for each `$\epsilon>0$` there is a `$\delta>0$` such that if,

`$$\begin{aligned}|x - x_{0} | & < \delta \text{ for all  } x \in \Re \text{ then } \nonumber \\|f(x) - f(x_{0})| & < \epsilon \nonumber \end{aligned}$$`

* Previously `$f(x_{0})$` was replaced with `$L$`
* Now: `$f(x)$` has to converge on itself at `$x_{0}$`
* **Continuity is more restrictive than limit**

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# Examples of continuity

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# Examples of continuity

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# Examples of continuity

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# Measuring incumbency advantage

* Incumbency advantage
* U.S. House of Representatives
* Win rate of incumbent parties `$>90\%$`
* Win rate of incumbent candidates `$\approx 90\%$` (conditional on running for election)
    * `$88\%$` probability of running for reelection
--
* Runners-up
    * `$3\%$` chance of winning the next election
    * Only `$20\%$` chance of running in the next election
* Is there an electoral advantage to incumbency?
* Can this be proven through observational study?

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# Ideal experiment

* Randomly assign incumbent parties in a district between Democrats and Republicans
* Keep all other factors constant
* Corresponding increase in Democratic electoral success in the next election would represent the overall electoral benefit due to being the incumbent party in the district
* Realistic?

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# Regression discontinuity design

* Dichotomous treatment is a deterministic function of a single, continuous covariate
* Treatment is assigned to those individuals whose score crosses a known threshold
* If you know the score, you can reverse-engineer the treatment assignment
* Assumes as-if random assignment in the local neighborhood around a probability of `$50\%$`

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# RDD + incumbency advantage

Source: [Lee, D. S. (2008). Randomized experiments from non-random selection in US House elections. *Journal of Econometrics*, 142(2), 675-697.](https://www.sciencedirect.com/science/article/pii/S0304407607001121)

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# Continuity and limits

* Let `$f: \Re \rightarrow \Re$` with `$x_{0} \in \Re$`
* Then `$f$` is continuous at `$x_{0}$` if and only if `$f$` has a limit at `$x_{0}$` and that `$\lim_{x \rightarrow x_{0} } f(x) = f(x_{0})$`
--

* Suppose `$f$` is continuous at `$x_{0}$`
    * This implies that `$|f(x) - f(x_0)| < \epsilon \, \text{for all} \, x \, \text{such that} \, |x - x_0 | < \delta$`
    * Definition of a **limit**, with `$L = f(x_{0})$`
--

* Suppose `$f$` has a limit at `$x_{0}$` and that limit is `$f(x_{0})$`
    * This implies that `$|f(x) - f(x_0)| < \epsilon \, \text{for all} \, x \, \text{such that} \, |x - x_0 | < \delta$`
    * This is the definition of **continuity**

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# Algebra of continuous functions

Suppose `$f:\Re \rightarrow \Re$` and `$g:\Re \rightarrow \Re$` are continuous at `$x_{0}$`. Then,

1. `$f(x) + g(x)$` is continuous at `$x_{0}$`
1. `$f(x) g(x)$` is continuous at `$x_{0}$`
1. If `$g(x_0) \neq 0$`, then `$\frac{f(x) } {g(x) }$` is continuous at `$x_{0}$`

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class: center, middle, inverse

# RECAP

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# Today's concepts:

* Sequences
  * arithmetic 
  * geometric
  * logs
  * convergence 
* Limits
  * continuity
  * what they tell us
  * how to find them
  
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# NEXT UP!

* DERIVATIVES!!!!!!!!
* Check in with any questions! (clipperton@uchicago.edu)

### TA TEAM!
* Bailey Meche baileymeche@uchicago.edu
* Gabe Reichman gabereichman@uchicago.edu
* Shirley Zhang xueyan1220@uchicago.edu
* Stanley Yi yijiaying@uchicago.edu

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class: center, middle, inverse
## (Students) Let’s get to know each other a bit more – Name, pronouns, subfield/research area, where you are currently, something fun/interesting about you and/or your hobbies