8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES
515
(i) Show that the Taylor series centered at 0 for e x converges to e x for every
nonnegative value of x.
(ii) Show that the Taylor series centered at 0 for e x converges to e x for every
negative value of x.
(iii) Explain why the Taylor series centered at 0 for e x converges to e x for
every real number x. Recall that we earlier showed that the Taylor series
centered at 0 for e x converges for all x, and we have now completed the
argument that the Taylor series for e x actually converges to e x for all x.
(c) Let P n (x) be the nth order Taylor polynomial for e x centered at 0. Find a value
of n so that P n (5) approximates e 5 correct to 8 decimal places.
⊳
Summary
In this section, we encountered the following important ideas:
• We can use Taylor polynomials to approximate complicated functions. This allows
us to approximate values of complicated functions using only addition, subtraction,
multiplication, and division of real numbers. The nth order Taylor polynomial centered
at x = a of a function f is
P n (x) = f (a) + f
′ (a)(x − a) +
f ′′ (a)
2!
(x − a)
2 + · · · +
f (n) (a)
n!
(x − a)
n
=
n
k=0
f (k) (a)
k!
(x − a)
k .
• The Taylor series centered at x = a for a function f is
∞
k=0
f (k) (a)
k!
(x − a)
k .
• The nth order Taylor polynomial centered at a for f is the nth partial sum of its
Taylor series centered at a. So the nth order Taylor polynomial for a function f is an
approximation to f on the interval where the Taylor series converges; for the values of
x for which the Taylor series converges to f we write
f (x) =
∞
k=0
f (k) (a)
k!
(x − a)
k .
• The Lagrange Error Bound shows us how to determine the accuracy in using a Taylor
polynomial to approximate a function. More specifically, if P n (x) is the nth order Taylor
515
(i) Show that the Taylor series centered at 0 for e x converges to e x for every
nonnegative value of x.
(ii) Show that the Taylor series centered at 0 for e x converges to e x for every
negative value of x.
(iii) Explain why the Taylor series centered at 0 for e x converges to e x for
every real number x. Recall that we earlier showed that the Taylor series
centered at 0 for e x converges for all x, and we have now completed the
argument that the Taylor series for e x actually converges to e x for all x.
(c) Let P n (x) be the nth order Taylor polynomial for e x centered at 0. Find a value
of n so that P n (5) approximates e 5 correct to 8 decimal places.
⊳
Summary
In this section, we encountered the following important ideas:
• We can use Taylor polynomials to approximate complicated functions. This allows
us to approximate values of complicated functions using only addition, subtraction,
multiplication, and division of real numbers. The nth order Taylor polynomial centered
at x = a of a function f is
P n (x) = f (a) + f
′ (a)(x − a) +
f ′′ (a)
2!
(x − a)
2 + · · · +
f (n) (a)
n!
(x − a)
n
=
n
k=0
f (k) (a)
k!
(x − a)
k .
• The Taylor series centered at x = a for a function f is
∞
k=0
f (k) (a)
k!
(x − a)
k .
• The nth order Taylor polynomial centered at a for f is the nth partial sum of its
Taylor series centered at a. So the nth order Taylor polynomial for a function f is an
approximation to f on the interval where the Taylor series converges; for the values of
x for which the Taylor series converges to f we write
f (x) =
∞
k=0
f (k) (a)
k!
(x − a)
k .
• The Lagrange Error Bound shows us how to determine the accuracy in using a Taylor
polynomial to approximate a function. More specifically, if P n (x) is the nth order Taylor