Column 1Column 2
F(x)f(x)
|x||x|x/2
Untitled

# Limit

For any x, for any neighborhood N of f(x), there exists a neighborhood of x whose image is contained in N.

Planning.

# Volume with cross-sections

## Worked examples

• $\int x^{1/ln(x)}dx$﻿

# Euler Method

But, diferential equations are

Thus

A good approx. step is $\Delta x=0.00001$﻿, meaning that we need 400,000 steps (high computational cost).

#### Step sizes

step sizeresult of Euler's methoderrorTitle
11638.6Untitled
0.2535.5319.07Untitled
0.145.269.34Untitled
0.0549.565.04Untitled
0.02551.982.62Untitled
0.012553.261.34Untitled

# Logistic models

https://es.wikipedia.org/wiki/Función_logística

# Summation and series

https://www.cis.rit.edu/class/simg716/series_you_should_know.pdf

https://en.wikipedia.org/wiki/List_of_mathematical_series

Definition. Given a sequence $a_1,a_2,...,a_n\in \mathbb{R}$﻿,

But you can start with another index while exists in the sequence.

• Examples.

## Product

Definition. Given a sequence $a_1,a_2,...,a_n\in \mathbb{R}$﻿,

# Bounding summations

## Worked examples

• $\sum_{k=1}^n(2k-1)$﻿
• $\text{Show that }\sum_{k=1}^n1/(2k-1)=ln(\sqrt n)+O(1)$﻿