Theory

Theory 1

Suppose that we have a power series function:

Consider the successive derivatives of :

When these functions are evaluated at , all terms with a positive -power become zero:

This last formula is the basis for Taylor and Maclaurin series:

Power series: Derivative-Coefficient Identity

This identity holds for a power series function which has a nonzero radius of convergence.

We can apply the identity in both directions:

• Know ? Calculate for any .
• Know ? Calculate for any .

---

Many functions can be ‘expressed’ or ‘represented’ near (i.e. for small enough ) as convergent power series. (This is true for almost all the functions encountered in pre-calculus and calculus.)

Such a power series representation is called a Taylor series. When , the Taylor series is also called the Maclaurin series.

One power series representation we have already studied:

---

Whenever a function has a power series (Taylor or Maclaurin), the Derivative-Coefficient Identity may be applied to calculate the coefficients of that series.

Conversely, sometimes a series can be interpreted as an evaluated power series coming from for some . If the closed form function format can be obtained for this power series, the total sum of the original series may be discovered by putting in the argument of the function.

Theory 2

Study these!

• Memorize all of these series!
• Recognize all of these series!
• Recognize all of these summation formulas!