Question

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.
||
:
+.
expand button
Transcribed Image Text: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. || : +.
2xcosh(x²).
expand button
Transcribed Image Text:2xcosh(x²).
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