Use the Maclaurin series representation of sinh(x) from the table to create a Maclaurin series of sinh(x^2). Next, differentiate it to create a Maclaurin series for 2xcosh(x^2) Table 11.51= 1 + x + x² + . . . + x* + . .8.+*, for |x < 1=1 x + x² -+(-1)'r* + • . .Σ(-1), for l지 < 11 + xx²e = 1 + x +2!.2.·+k!for x < ∞k!k=0(-1)*x²k+1(2k + 1)!- -1)*x*+1(2k + 1)!sin x x -3!for x < 05!k=0(-1)' x²*(2k)!2k(-1)*x*cos x = 1 -2!for x< ∞4!k=0(2k)!(-1)**'x*(-1)k+l,kIn (1 + x) = x -23.for -1 < x < 1k.k=1ナイ+· • · +.3.–In (1 – x) = x +for -1 < x < 1k=1(-1)*x²+1+. .(-1) x3.(-1)*x²*+1nx= .3.-1tanfor |x| < 12k + 12k + 15..32k+12k+1sinh x = x +3!+· · ·+5!for x < ∞%3D(2k+ 1)!k=0(2k + 1)!'.22k00cosh x = 1+2!for x<∞4!(2k)!(2k)!'p(p - 1)(p – 2)- - (p – k + 1)(1 + x) = E("), for |x| < 1 and()%3D1%3D%3Dk!8.8.||:+. 2xcosh(x²).

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Answered: Use the Maclaurin series representation…

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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Use the Maclaurin series representation of sinh(x) from the table to create a Maclaurin series of sinh(x^2). Next, differentiate it to create a Maclaurin series for 2xcosh(x^2)

Table 11.5
1
= 1 + x + x² + . . . + x* + . .
8.
+*, for |x < 1
=1 x + x² -
+(-1)'r* + • . .
Σ(-1), for l지 < 1
1 + x
x²
e = 1 + x +
2!
.2.
·+
k!
for x < ∞
k!
k=0
(-1)*x²k+1
(2k + 1)!
- -1)*x*+1
(2k + 1)!
sin x x -
3!
for x < 0
5!
k=0
(-1)' x²*
(2k)!
2k
(-1)*x*
cos x = 1 -
2!
for x< ∞
4!
k=0
(2k)!
(-1)**'x*
(-1)k+l,k
In (1 + x) = x -
2
3.
for -1 < x < 1
k.
k=1
ナイ
+· • · +
.3.
–In (1 – x) = x +
for -1 < x < 1
k=1
(-1)*x²+1
+. .
(-1) x
3.
(-1)*x²*+1
nx= .
3.
-1
tan
for |x| < 1
2k + 1
2k + 1
5.
.3
2k+1
2k+1
sinh x = x +
3!
+· · ·+
5!
for x < ∞
%3D
(2k+ 1)!
k=0(2k + 1)!'
.2
2k
00
cosh x = 1+
2!
for x<∞
4!
(2k)!
(2k)!'
p(p - 1)(p – 2)- - (p – k + 1)
(1 + x) = E(")
, for |x| < 1 and
()
%3D
1
%3D
%3D
k!
8.
8.
||
:
+.

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