MathDB
Problems
Contests
National and Regional Contests
Kyrgyzstan Contests
Kyrgyzstan National Olympiad
2011 Kyrgyzstan National Olympiad
7
Difference problem(KgNM2011)
Difference problem(KgNM2011)
Source:
April 20, 2011
induction
function
continued fraction
algebra unsolved
algebra
Problem Statement
Given that
g
(
n
)
=
1
2
+
1
3
+
1
.
.
.
+
1
n
−
1
g(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1}}}}}}}}
g
(
n
)
=
2
+
3
+
...
+
n
−
1
1
1
1
1
and
k
(
n
)
=
1
2
+
1
3
+
1
.
.
.
+
1
n
−
1
+
1
n
k(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1 + \frac{1}{n}}}}}}}}}
k
(
n
)
=
2
+
3
+
...
+
n
−
1
+
n
1
1
1
1
1
, for natural
n
n
n
. Prove that
∣
g
(
n
)
−
k
(
n
)
∣
≤
1
(
n
−
1
)
!
n
!
\left| {g(n) - k(n)} \right| \le \frac{1}{{(n - 1)!n!}}
∣
g
(
n
)
−
k
(
n
)
∣
≤
(
n
−
1
)!
n
!
1
.
Back to Problems
View on AoPS