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Mathematics
Mathematics
Problem
If
f
(
x
)
=
x
+
x
2
1
!
+
x
3
2
!
+
⋯
+
x
n
(
n
−
1
)
!
f(x) = x + \frac{x^2}{1!} + \frac{x^3}{2!} + \dots + \frac{x^n}{(n-1)!}
f
(
x
)
=
x
+
1
!
x
2
+
2
!
x
3
+
⋯
+
(
n
−
1
)!
x
n
, then
f
(
0
)
+
f
′
(
0
)
+
f
′
′
(
0
)
+
⋯
+
f
(
n
)
(
0
)
f(0) + f'(0) + f''(0) + \dots + f^{(n)}(0)
f
(
0
)
+
f
′
(
0
)
+
f
′′
(
0
)
+
⋯
+
f
(
n
)
(
0
)
is equal to a.
n
(
n
+
1
)
2
\frac{n(n+1)}{2}
2
n
(
n
+
1
)
b.
(
n
2
+
1
)
2
\frac{(n^2+1)}{2}
2
(
n
2
+
1
)
c.
(
n
(
n
+
1
)
2
)
2
\left(\frac{n(n+1)}{2}\right)^2
(
2
n
(
n
+
1
)
)
2
d.
n
(
n
+
1
)
(
2
n
+
1
)
6
\frac{n(n+1)(2n+1)}{6}
6
n
(
n
+
1
)
(
2
n
+
1
)
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