$$ \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\EE}{\mathbb{E}} \newcommand{\HH}{\mathbb{H}} \newcommand{\FF}{\mathbb{F}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\length}{\operatorname{length}} \newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}} \newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}} \newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}} \newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}} \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}} \newcommand{\partitions}[1]{\mathcal{P}_{#1}} \newcommand{\erf}{\operatorname{erf}} \newcommand{\ihat}{\hat{\imath}} \newcommand{\jhat}{\hat{\jmath}} \newcommand{\khat}{\hat{k}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \newcommand{\smat}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)} $$

Study II

Write a study guide for yourself in preparation for Midterm II.
This is an optional assignment - if you are trying for the “Hold Me Accountable” grading scale (where I weight your exams a bit less in total points, by giving you points for studying/reflecting on them) then you should do this, and turn this in. If you do not want to write a study guide that is fine, it will not hurt you if you are going for the “I got this” grading scheme, where I’ll instead weight your exams higher and not ask for these smaller assignments along the way.

The instructions are the same as Study I: you can write this however helps you most, but only submissions showing clear effort will be awarded “hold me accountable” points. Below is a brief listing of the topics we have covered since Midterm I, which will be the focus of Midterm II.

Sequences

You need to be able to effectively use the previous material on sequences in your calculations / proofs / arguments, including the definition of convergence, and limit laws / limit inequalities / squeeze theorem. The new material that may appear on the exam is below:

  • The Monotone Convergence Theorem: its proof, and using it to calculate limits.
  • Subsequences: breaking a sequence into subsequences to prove convergence
  • The Bolzano Weierstrass Theorem
  • The definitions of Limsup and Liminf
  • Cauchy sequences: definition + theorem that cauchy is equivalent to convergence
  • Contraction Mapping Theorem (you are not responsible for the proof, but should be able to use the theorem)

Series

  • Definition of infinite series and products as limits
  • Examples we can compute directly: telescoping and geometric series
  • Absolute convergence
  • Divergence test and alternating series test
  • The comparison theorem
  • Comparing with a geometric series: the root and ratio tests
  • Power series, and using root/ratio test to find convergence
  • Dominated Convergence (you are not responsible for the proof but should be able to use the theorem)

Continuity

  • Definition in terms of \(\epsilon-\delta\)
  • Proving continuity with sequences
  • The “Continuity Laws”
  • Definition of Function Limits
  • Continuity of Power Series
  • Continuity on Dense Sets
  • Extreme and Intermediate Value Theorems (know their statements; I won’t ask about their proofs)