Introduction to Taylor Series Why are we looking at power series? If we reverse the equation for the geometric series: 1 1 + x + x2 + x3 + · · · = 1−x we get a description of 1−1 x in terms of a series. In fact, we can represent all of the functions we’ve encountered in this course in terms of series. The technique is similar to the use of a decimal expansion to represent 1/3 √ or 2. When we describe a function like ex or arctan x in terms of a series we can approximate and manipulate those functions as easily as we do polynomial functions. Rules for convergent power series What sorts of manipulations might we want to perform? Addition, multiplica tion, division, substitution (composition), integration and diﬀerentiation. The rules for manipulation of power series are essentially the same as those for ma nipulating polynomials! � d f (x) + g(x), f (x) · g(x), f (g(x)), f (x)/g(x), f (x), f (x)dx dx We can do all of these with power series; in this class integration and diﬀeren tiation will be the most interesting manipulations. We take the derivative of a power series just as we do for polynomials: d (a0 + a1 x + a2 x2 + a3 x3 + · · ·) = a1 + 2a2 x + 3a3 x2 + · · · dx Similarly, the formula for the integral of a power series is: � a1 x2 a2 x3 a3 x4 (a0 + a1 x + a2 x2 + a3 x3 + · · ·)dx = c + a0 x + + + + ··· 2 3 4 Here the arbitrary constant c takes the place of the constant term in the new series. Question: Is that a series or a polynomial? Answer: It’s a polynomial if it ends; if it goes on inﬁnitely far then it’s a series. Question: You can add up terms of x in a series? Answer: When we introduced series we described them as inﬁnite sums of numbers. At the start of this class we rewrote the geometric series using the variable x in place of the “constant value” a. When we plug in a value for x we get a sum of inﬁnitely many numbers, so as long as we remember that x is a placeholder for a numerical value there’s no problem. In other words, what we’re working with here are functions of x. These func tions are deﬁned for values of x inside the radius of convergence and undeﬁned √ for values of x that are too large, just as the function f (x) = x is deﬁned for positive values of x and undeﬁned for x < 0. Power series are inﬁnite sums of powers of x, with coeﬃcients. People also study and use series that are inﬁnite sums of sines and cosines and lots of other series, but we’re only going to study power series here.