## Introduction to Power Series (15 minutes)

• SWBQ: Write down something you remember about power series.
• In physics we use the term power series to refer to both Taylor and MacLaurin and Laurent series.
• Power series are a valuable way to approximate a function at a point, and are a strong tool for physics sense-making.
• While a function might not be integrable, the power series of the function can be integrated term by term.
• The terms and coefficients are labeled as 0th, 1st, 2nd, … order, referring to the exponent.
• Expanding to $n$th order means that all terms up to $z^n$ should be calculated.
• Using $z$ and an arbitrary $z_0$, derive the formula for the coefficients (most students know this formula, but they don't remember the derivation).

## Properties of Power Series (15 minutes)

• The power series for a function about a point is unique
• This is a license to do anything you want! If you get an answer, it's the correct answer (so long as your algebra is correct).
• There are a whole bunch of theorems and properties posted on the website, most importantly:

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