Chapter 3

I. Complete the following 3 problems:

• Exercise 3-36, p. 118
• Exercise 3-40, p. 119
• Sample Exam Problem 3-16, p. 132

II. Read and digest Section 3.7 (pp. 126-128) on probability generating functions, and create Anki cards summarizing the main results about probability generating functions.

Pointer: If you would like to typeset any of the equations for your cards directly into Anki using LaTeX and are unfamiliar with LaTeX, please see me for a quick tutorial. You might also find the tutorial here useful.

III. Evaluate the following sum or series:

1. \(\sum\limits_{n = a}^{b} n,\) where \(a, b \in \mathbb{Z}, a < b\)
2. \(\sum\limits_{n = 1}^{\infty} (1-p)^{n-1} p, p \in [0, 1]\)

IV. Read about power series and Taylor series in your favorite calculus textbook. Any calculus textbook will do, and they all have a chapter with a title along the lines of "infinite sequences and series." If you do not have a favorite calculus textbook, here are some free options:

• Active Calculus
• Apex Calculus
• Community Calculus

Write one page of notes on Taylor series, explaining how to derive a Taylor series from a given differentiable function. Then find the Taylor series expansion of the following functions

1. \(e^{x}\)
2. \(\sin x\)
3. \(\frac{x}{1 - x}\)

V. Complete two problems from Problem Solving Session 2 and one problem from Problem Solving Session 1 that you did not solve during the problem solving session and that were not presented by the student presenters.