The Maclaurin series is a template that allows you to express many other functions as power series. It is the source of formulas for expressing both sin x and cos x as infinite series. Without further ado, here it is:The notation f(n) means “the nth derivative of f.” This becomes clearer in the expanded version of the Maclaurin series:The Maclaurin series allows you to express functions as power series by following these steps: Find the first few derivatives of the function until you recognize a pattern. Substitute 0 for x into each of these derivatives. Plug these values, term by term, into the formula for the Maclaurin series. If possible, express the series in sigma notation. For example, suppose that you want to find the Maclaurin series for ex. Find the first few derivatives of ex until you recognize a pattern: Substitute 0 for x into each of these derivatives. Plug these values, term by term, into the formula for the Maclaurin series: If possible, express the series in sigma nota...
