Generating Functions

gives a single object representing the entire sequence. This is particularly useful when the series is the Taylor series of a familiar function. An important example is ak D 1 for all k 0. The series A.x/ is a geometric series that is known to converge to .1 x/ . Other examples are obtained by performing simple operations on this one. When you met infinite series in Calculus, the Taylor series was given star billing. It allowed the coefficients of a series representation of a function to be expressed in terms of the values of the derivatives of the function at a single point (which will be x D 0 in all examples in this course). The statement is just what you expect: if you formally find .d=dx/A.x/ using the series, the terms of degree less than n become zero, the terms of degree greater than n retain at least one factor of x, so they become zero at x D 0 and anx becomes the constant annŠ, and Taylor’s Theorem says nothing more than annŠ D A.0/. Because of this, differentiation and integration will play an important role in constructing new series from old, but there are places where a literal use of Taylor’s formula is more complicated than other methods. Taylor’s formula, with an error term, still plays a role. It shows that the series has the expected limit. Note that Taylor’s formula produces the coefficients of the series using values of successive derivatives at zero; no simpler relation can be expected to give correct answers. In particular, the A.k/ (which were observed in some answers on the second midterm, offer no shortcut to finding the ak .