Fermi
·
Concept Library
·
Courses
·
Books
·
Problems
·
Simulations
·
Math Wiki
·
High School
Fermi
›
Problems
Mathematics
Mathematics
Problem
Let
f
(
x
)
f(x)
f
(
x
)
be a differentiable function satisfying
f
(
x
)
+
f
(
x
+
1
2
)
=
1
,
∀
x
∈
R
f(x) + f\left(x + \frac{1}{2}\right) = 1, \forall x \in \mathbb{R}
f
(
x
)
+
f
(
x
+
2
1
)
=
1
,
∀
x
∈
R
and
g
(
x
)
=
∫
0
x
f
(
t
)
d
t
g(x) = \int_{0}^{x} f(t) dt
g
(
x
)
=
∫
0
x
f
(
t
)
d
t
. If
g
(
1
)
=
1
g(1) = 1
g
(
1
)
=
1
, then the value of
∑
n
=
2
∞
(
8
∑
k
=
1
n
(
g
(
x
+
k
2
)
−
g
(
x
+
k
)
)
)
\sum_{n=2}^{\infty} \left( \frac{8}{\sum_{k=1}^{n} (g(x+k^2) - g(x+k))} \right)
∑
n
=
2
∞
(
∑
k
=
1
n
(
g
(
x
+
k
2
)
−
g
(
x
+
k
))
8
)
is.
Browse the problem library