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How to find the coefficients of a power series (taylor or maclaurin) for a given function using the function's derivatives at a point. It also discusses the concept of taylor polynomials and the remainder, which determines how well the polynomial approximates the function. An example of the maclaurian series for e^x and demonstrates the application of taylor's inequality for the remainder.
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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.
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.