6.5 Taylor's Theorem 339SOME WORD S OF CAUTIONTaylor polynomials Tn ( x) about a of a function f are most reliable as
approximations to f(x) when f has derivatives of all orders in a neighborhood
of a and lim Rn.(x) = 0 for all x in this neighborhood. In the examples we
n->oo
have seen, this was not a severe limitation. In fact, in Example 6.5. 13 we saw
that for the function f ( x) = ex, lim Rn ( x) = 0 for all values of x. In Exercise
n->oo
13 you will prove that the same is true for the sine and cosine functions. But
there are functions for which lim Rn(x) is not zero or does not even exist.
n->oo
For an extreme example, see Exercise 6.6.16. In this example, Tn(x) about
0 is the constant zero function for all values of n, and thus the sequence {Tn ( x)}
does not converge to the given function f(x) for any x =I-0.
Finally, practical concerns such as determining whether Tn(x) will approx-
imate f(x) to within a prescribed degree of accuracy, how close x must be to
a and how large n must be to guarantee that accuracy, and what computa-
tional procedures are most efficient, are left to specialized applied mathematics
courses such as numerical analysis.
EXERCISE SET 6.5l. Expand the terms of T 4 (x) obtained in Example 6.5.4, and show that
T4(x) = f(x).- Prove Theorem 6.5.7 for n = 4.
- Find the Taylor polynomials T 2 n+ 1 (x) and T2n(x) about 0 for the function
f(x) = sinx. - Use Taylor's theorem and the result of Exercise 3 to prove that l::/x E (0, 7r),
x3 x3 x5
x - - < sin x < x - - + -.
3! 3! 5!
[Hint: Calculate T 3 (x) about 0, and use R3(x) to obtain the first inequal-
ity; use T 5 (x) and R 5 (x) to obtain the other.] - Find the Taylor polynomials T2n(x) and T2n+1(x) about 0 for f(x) =
cosx. - Use the result of Exercise 5 to prove that l::/x E (0, 7r),
x2 x4 x2 x4 x6
1 - 2! + 4! > cos x > 1 - 2! + 4! - 6!.
[Hint: Calculate T 5 (x) about 0, and use R 5 (x) to obtain the first inequal-
ity; use T 7 (x) and R 7 (x) to obtain the other.] - Find the sixth Taylor polynomial T5(x) for the function f(x) = ,/X about
l. Also, write the formula for the Lagrange form of R5(x).