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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:
