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Math 113 Lecture
§11.10: Taylor and Maclaurin Series, Part I
Taylor’s Formula for the Coefficients. Last time we saw how to use the geometric
series to express certain kinds of functions as power series. We now learn how to express many functions as power series. We start by assuming that we have a power series with a positive radius of convergence:
f (x) =
∑^ ∞
n=
cn(x − a)n, |x − a| < R.
We learned last time that such an f is infinitely differentiable. If we know what f (x) is, how do we find the coefficients c 0 , c 1 , c 2 ,...? Well, if we evaluate the power series at its center, we get
f (a) =
∑^ ∞
n=
cn(a − a)n^ =
∑^ ∞
n=
cn 0 n^ = c 0 + 0c 1 + 0c 2 + · · ·.
This gives c 0 = f (a). How do we find c 1? Well, we evalutate f ′(x) at the center:
f ′(a) =
∑^ ∞
n=
ncn(a − a)n−^1 = 1c 1 + 0c 2 + 0c 3 + · · ·.
So we have c 1 = f ′(a). It now stands to reason that c 2 is somehow related to the second derivative of f :
f ′′(a) =
∑^ ∞
n=
n(n − 1)cnxn−^2 = 1 · 2 · c 2 + 0c 3 + 0c 4 + · · ·.
Thus we have c 2 = f^
′′(a) 1 · 2 =^
f (2)(a) 2!. By f (n)(a) we mean the nth^ derivative of f evaluated at a. Continuing the above pattern gives c 3 :
f (3)(a) =
∑^ ∞
n=
n(n − 1)(n − 2)cn(a − a)n−^3 = 1 · 2 · 3 · c 3 + 0c 4 + 0c 5 + · · ·.
This gives
c 3 = f^
(3)(a) 3!.
By now you should be able to guess the formula for the value of the nth^ coefficient in the power series for f :
cn = f^
(n)(a) n!. Putting this into the power series for f gives
f (x) =
∑^ ∞
n=
f (n)(a) n! (x^ −^ a)
n.
This power series is a called the Taylor series of the function f about a. This power series is called a Maclaurin series if a = 0. A word of warning on this Taylor series: we have shown that if f can be represented by a convergent power series (i.e., R > 0), then f is equal to its Taylor series. There are infinitely differentiable functions which are not equal to their Taylor series about a, such as
f (x) =
exp(− 1 /x^2 ) x 6 = 0, 0 x = 0, about a = 0.
Taylor Polynomials and the Remainder. We now consider the question of whether
an infinitely differentiable function f is equal to its Taylor series or not. This requires an investigation of the sequence of partial sums for the Taylor series and their remainders. The nth^ partial sum of a Taylor series for f about a is called the nth-degree Taylor Polynomial of f at a:
Tn(x) =
∑^ n k=
f (k)(a) k! (x^ −^ a)
k.
The limit of the sequence of nth-degree Taylor Polynomials is the sum of the Taylor series:
∑^ ∞ n=
f (n)(a) n! (x^ −^ a)
n (^) = lim n→∞ Tn(x).
How good of an approximation Tn(x) is to f (x) is determined by the remainder:
Rn(x) = f (x) − Tn(x), i.e., f (x) = Tn(x) + Rn(x).
For Tn(x) to converge to f (x), i.e., for the Taylor series of f to be equal to f , requires that
nlim→∞ Rn(x) = 0. All of this discussion is on the interval |x − a| < R, where R is the radius of convergence of the Taylor series for the infinitely differentiable function f about a.
