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:
    1. FIXME

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