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Mathematics
Mathematics
Problem
Let
f
f
f
be analytic at
x
=
0
x=0
x
=
0
with Maclaurin series
f
(
x
)
=
∑
k
=
0
∞
a
k
x
k
f(x)=\sum_{k=0}^{\infty}a_k x^{k}
f
(
x
)
=
∑
k
=
0
∞
a
k
x
k
. Suppose
(
1
+
x
+
x
2
2
+
x
3
6
+
⋯
)
f
(
x
)
=
1
+
x
2
+
2
3
x
3
+
⋯
\bigl(1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\cdots\bigr)\,f(x)=1+x^{2}+\frac{2}{3}x^{3}+\cdots
(
1
+
x
+
2
x
2
+
6
x
3
+
⋯
)
f
(
x
)
=
1
+
x
2
+
3
2
x
3
+
⋯
for all
x
x
x
near
0
0
0
. What is
f
′
′
′
(
0
)
f'''(0)
f
′′′
(
0
)
? a)
−
6
-6
−
6
b)
−
3
-3
−
3
c)
−
1
2
-\dfrac12
−
2
1
d)
3
3
3
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