Table 11.5 asserts, without proof, that in several cases, the Taylor series for f converges to f at the endpoints of the interval of convergence. Proving convergence at the endpoints generally requires advanced techniques. It may also be done using the following theorem: Suppose the Taylor series for f entered at 0 converges to f on the interval (-R, R). If the series converges at x = R, then it converges to lim f(x). Table 11.5 = 1 + x + x? + + x* Σxt. for 지 < 1 k=0 1- x + x? - ... + (-1)*r* + - E(-1)*r*, for |x| < 1 k=0 x2 et = 1 +x + 2! Σ for x| < 0 k! k!' k=0 (-1)*x*+1 Eo (2k + 1)! х3 sin x = x - 3! (-1)*x*+1 for |x| < 0 If the series converges at x = -R, then it 5! (2k + 1)! converges to lim f(x). x--R (-1)*x* Σ (2k)! Forexample, this theorem would x² cos x = 1 (-1)*x* for x| < 0 allow us to conclude that the series for (2k)! In (1 + x) converges to In 2 at x = 1. (-1)*+l,* Σ x2 x3 (-1)*+1x* In (1 + x) = for -1
Table 11.5 asserts, without proof, that in several cases, the Taylor series for f converges to f at the endpoints of the interval of convergence. Proving convergence at the endpoints generally requires advanced techniques. It may also be done using the following theorem: Suppose the Taylor series for f entered at 0 converges to f on the interval (-R, R). If the series converges at x = R, then it converges to lim f(x). Table 11.5 = 1 + x + x? + + x* Σxt. for 지 < 1 k=0 1- x + x? - ... + (-1)*r* + - E(-1)*r*, for |x| < 1 k=0 x2 et = 1 +x + 2! Σ for x| < 0 k! k!' k=0 (-1)*x*+1 Eo (2k + 1)! х3 sin x = x - 3! (-1)*x*+1 for |x| < 0 If the series converges at x = -R, then it 5! (2k + 1)! converges to lim f(x). x--R (-1)*x* Σ (2k)! Forexample, this theorem would x² cos x = 1 (-1)*x* for x| < 0 allow us to conclude that the series for (2k)! In (1 + x) converges to In 2 at x = 1. (-1)*+l,* Σ x2 x3 (-1)*+1x* In (1 + x) = for -1
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.
Transcribed Image Text:Table 11.5 asserts, without proof, that
in several cases, the Taylor series for
f converges to f at the endpoints of
the interval of convergence. Proving
convergence at the endpoints generally
requires advanced techniques. It may also
be done using the following theorem:
Suppose the Taylor series for f
entered at 0 converges to f on the
interval (-R, R). If the series converges
at x = R, then it converges to lim f(x).
Table 11.5
= 1 + x + x? +
+ x*
Σxt. for 지 < 1
k=0
1- x + x? - ... + (-1)*r* + -
E(-1)*r*, for |x| < 1
k=0
x2
et = 1 +x +
2!
Σ
for x| < 0
k!
k!'
k=0
(-1)*x*+1
Eo (2k + 1)!
х3
sin x = x -
3!
(-1)*x*+1
for |x| < 0
If the series converges at x = -R, then it
5!
(2k + 1)!
converges to lim f(x).
x--R
(-1)*x*
Σ
(2k)!
Forexample, this theorem would
x²
cos x = 1
(-1)*x*
for x| < 0
allow us to conclude that the series for
(2k)!
In (1 + x) converges to In 2 at x = 1.
(-1)*+l,*
Σ
x2
x3
(-1)*+1x*
In (1 + x) =
for -1 <xs1
k=1
x x3
x*
-In (1 - x) = x +
for -1 sx <1
... %3D
k=1
(-1)*x*+1
Σ
(-1)* x*+1
tanx = x
for |x| s 1
2k + 1
2k + 1
x*+1
x2k+1
Σ
for |x| < 0
sinh x = x +
5!
(2k + 1)!
(2k + 1)!'
x*
cosh x
for |x| < 0
+..
2!
4!
(2k)!
(2k)!"
moted in Theorem 11.6 the hinoial
erieay converge to (1 at
+1 denendine on the value of a
(3)
p(p – 1)(p – 2) - (p – k + 1)
(1 + x)P = E()x
2(3) -
r*, for x| < 1 and
k!
Transcribed Image Text:1
37.
1- 2x
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