Consider f(x) f(x) (a) Use a Maclaurin series from Table 1 in section 11.10 of the text (or do an internet search for 'table of Maclaurin series') to find the Maclaurin series of f(x). = ∞ Σ n=1 XE 1 - cos(x³) x3 (b) Find the interval on which the series in (a) converges to f(x). (Note that f(x) is not defined for all x.) State the answer using interval notation. Note: Input U, infinity, and -infinity for union, ∞, and -∞, respectively.
Consider f(x) f(x) (a) Use a Maclaurin series from Table 1 in section 11.10 of the text (or do an internet search for 'table of Maclaurin series') to find the Maclaurin series of f(x). = ∞ Σ n=1 XE 1 - cos(x³) x3 (b) Find the interval on which the series in (a) converges to f(x). (Note that f(x) is not defined for all x.) State the answer using interval notation. Note: Input U, infinity, and -infinity for union, ∞, and -∞, respectively.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Table 1
Important Maclaurin Series
and Their Radii of
Convergence
ex
1
- X
=
sin x =
=
Σ
n=0_n!
COS X =
Σx² = 1 + x + x² + x³ +
n=0
Σ(-1)"
n=0
Σ(-1)"
n=0
00
1 + +
n=0
X x² X
+
1! 2! 3!
tan ¹x = Σ(-1)″
x²n+1
(2n + 1)!
n=1
x2n
(2n)!
x²n+1
2n + 1
= X
= X
xn
In(1 + x) = Σ(−1)n-1 = X
n
2!
(1 + x)* = Σ x"= 1+ kx +
Σ( ^) +-
n=0
n
+
x³ x5
+
3! 5!
+
x³
3
X
2
X
4! 6!
x5
g g
5
7
+
+
X
3
k(k − 1)
2!
7!
X
4
-x² +
+
+
k(k − 1)(k − 2)
3!
-x³ +
R = 1
R = ∞
R = ∞
R = ∞
R = 1
R = 1
R = 1
Transcribed Image Text:Consider f(x) =
=
(a) Use a Maclaurin series from Table 1 in section 11.10 of the text (or do an internet search for 'table of Maclaurin series') to find the Maclaurin series of
f(x).
f(x) =
∞
x E
1 - cos(2³)
x³
n=1
(b) Find the interval on which the series in (a) converges to f(x). (Note that f(x) is not defined for all x.) State the answer using interval notation.
Note: Input U, infinity, and -infinity for union, ∞, and -∞, respectively.
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