8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES
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is always an interval centered at x = a. For this reason, the set on which a Taylor series
converges is called the interval of convergence. Half the length of the interval of convergence
is called the radius of convergence. If the interval of convergence of a Taylor series is infinite,
then we say that the radius of convergence is infinite.
Activity 8.25.
(a) Use the Ratio Test to explicitly determine the interval of convergence of the
Taylor series for f (x) =
1
1−x centered at x = 0.
(b) Use the Ratio Test to explicitly determine the interval of convergence of the
Taylor series for f (x) = cos(x) centered at x = 0.
(c) Use the Ratio Test to explicitly determine the interval of convergence of the
Taylor series for f (x) = sin(x) centered at x = 0.
⊳
The Ratio Test tells us how we can determine the set of x values for which a Taylor
series converges absolutely. However, just because a Taylor series for a function f
converges, we cannot be certain that the Taylor series actually converges to f (x) on its
interval of convergence. To show why and where a Taylor series does in fact converge to
the function f , we next consider the error that is present in Taylor polynomials.
Error Approximations for Taylor Polynomials
We now know how to find Taylor polynomials for functions such as sin(x), as well as
how to determine the interval of convergence of the corresponding Taylor series. We
next develop an error bound that will tell us how well an nth order Taylor polynomial
P n (x) approximates its generating function f (x). This error bound will also allow us
to determine whether a Taylor series on its interval of convergence actually equals the
function f from which the Taylor series is derived. Finally, we will be able to use the error
bound to determine the order of the Taylor polynomial P n (x) for a function f that we
need to ensure that P n (x) approximates f (x) to any desired degree of accuracy.
In all of this, we need to compare P n (x) to f (x). For this argument, we assume throughout that we center our approximations at 0 (a similar argument holds for approximations
centered at a). We define the exact error, E n (x), that results from approximating f (x) with
P n (x) by
E n (x) = f (x) − P n (x).
We are particularly interested in |E n (x)|, the distance between P n and f . Note that since
P
(k)
n (0) = f
(k) (0)
for 0 ≤ k ≤ n, we know that
E
(k)
n (0) = 0
511
is always an interval centered at x = a. For this reason, the set on which a Taylor series
converges is called the interval of convergence. Half the length of the interval of convergence
is called the radius of convergence. If the interval of convergence of a Taylor series is infinite,
then we say that the radius of convergence is infinite.
Activity 8.25.
(a) Use the Ratio Test to explicitly determine the interval of convergence of the
Taylor series for f (x) =
1
1−x centered at x = 0.
(b) Use the Ratio Test to explicitly determine the interval of convergence of the
Taylor series for f (x) = cos(x) centered at x = 0.
(c) Use the Ratio Test to explicitly determine the interval of convergence of the
Taylor series for f (x) = sin(x) centered at x = 0.
⊳
The Ratio Test tells us how we can determine the set of x values for which a Taylor
series converges absolutely. However, just because a Taylor series for a function f
converges, we cannot be certain that the Taylor series actually converges to f (x) on its
interval of convergence. To show why and where a Taylor series does in fact converge to
the function f , we next consider the error that is present in Taylor polynomials.
Error Approximations for Taylor Polynomials
We now know how to find Taylor polynomials for functions such as sin(x), as well as
how to determine the interval of convergence of the corresponding Taylor series. We
next develop an error bound that will tell us how well an nth order Taylor polynomial
P n (x) approximates its generating function f (x). This error bound will also allow us
to determine whether a Taylor series on its interval of convergence actually equals the
function f from which the Taylor series is derived. Finally, we will be able to use the error
bound to determine the order of the Taylor polynomial P n (x) for a function f that we
need to ensure that P n (x) approximates f (x) to any desired degree of accuracy.
In all of this, we need to compare P n (x) to f (x). For this argument, we assume throughout that we center our approximations at 0 (a similar argument holds for approximations
centered at a). We define the exact error, E n (x), that results from approximating f (x) with
P n (x) by
E n (x) = f (x) − P n (x).
We are particularly interested in |E n (x)|, the distance between P n and f . Note that since
P
(k)
n (0) = f
(k) (0)
for 0 ≤ k ≤ n, we know that
E
(k)
n (0) = 0