02 Numbers Sequences Series Ipynb Colab

There was an error while loading. Please reload this page. In this lab, we will use SageMath to determine the convergence or divergence of a sequence of numbers and of infinite series. Consider the sequence {cos⁡(nπ)arctan⁡(n)}n=1∞\{\cos(n\pi)\arctan(n)\}_{n=1}^\infty{cos(nπ)arctan(n)}n=1∞​. We start by determining the first 10 terms of this sequence. We can do this in SageMath by letting an=cos⁡(nπ)arctan⁡(n)a_n = \cos(n\pi)\arctan(n)an​=cos(nπ)arctan(n) and then using a for\textbf{for}for loop and the range\textbf{range}range command.

We can get a better idea of what these numbers are by using the round\textbf{round}round command. Note that the terms of the sequence do not appear to approach a specific number. We can better tell what is happening by plotting the first 100 or so terms of the sequence. We can plot a point in SageMath by using the point\textbf{point}point command along with the show\textbf{show}show command. To plot multiple points on the same plot, we will store the points in a list and then show the list. SageMath does not allow us to plug the list directly into the show\textbf{show}show command.

Instead, we must input the sum of the elements in the list. From the graph, we see that the odd terms are approaching a specific value, namely −π2-\frac{\pi}{2}−2π​, and the even terms are approaching a specfic value, namely π2.\frac{\pi}{2}.2π​. However, since these values are different, the sequence diverges. Please note that this tutorial requires the user to have a basic understanding of the options available in Jupyter. If you are not familiar with Jupyter, we recommend exploring other tutorials in section to get started: The .ipynb file format stands for IPython Notebook, which was the original name of Jupyter Notebook.

This file format allows users to create and share interactive documents that contain: Notebooks can be used for a wide range of purposes, including data exploration, data visualization, machine learning, and scientific research. Notebooks consist of a series of cells, which can be either code cells or markdown/text cells. Code cells contain executable code in the programming language of your choice (e.g. Python, R, Julia, etc.). The code cells can be executed in the notebook, allowing you to see the output of your code and visualize your data in real time.

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