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Mathematics
Problem
Define
g
(
x
)
=
∑
n
=
0
∞
(
−
1
)
n
x
2
n
+
2
(
2
n
+
1
)
(
2
n
+
2
)
g(x)=\displaystyle\sum_{n=0}^{\infty}\dfrac{(-1)^{n}x^{2n+2}}{(2n+1)(2n+2)}
g
(
x
)
=
n
=
0
∑
∞
(
2
n
+
1
)
(
2
n
+
2
)
(
−
1
)
n
x
2
n
+
2
for
∣
x
∣
≤
1
|x|\le 1
∣
x
∣
≤
1
. Which of the following expressions is equal to
g
(
x
)
g(x)
g
(
x
)
for
∣
x
∣
≤
1
|x|\le 1
∣
x
∣
≤
1
? a)
1
2
ln
(
1
+
x
2
)
\dfrac{1}{2}\ln\bigl(1+x^{2}\bigr)
2
1
ln
(
1
+
x
2
)
b)
x
arctan
x
−
1
2
ln
(
1
+
x
2
)
x\arctan x-\dfrac{1}{2}\ln\bigl(1+x^{2}\bigr)
x
arctan
x
−
2
1
ln
(
1
+
x
2
)
c)
x
arctan
x
+
1
2
ln
(
1
+
x
2
)
x\arctan x+\dfrac{1}{2}\ln\bigl(1+x^{2}\bigr)
x
arctan
x
+
2
1
ln
(
1
+
x
2
)
d)
arctan
x
\arctan x
arctan
x
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