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Greece National Olympiad
1988 Greece National Olympiad
4
lim sum 1/ (a_i ... a_n) when a_{n+1}= a^2_{n}-2
lim sum 1/ (a_i ... a_n) when a_{n+1}= a^2_{n}-2
Source: 1988 Greece MO Grade XII p4
September 7, 2024
Sequences
limit
real analysis
Problem Statement
Let
a
1
=
5
a_1=5
a
1
=
5
and
a
n
+
1
=
a
n
2
−
2
a_{n+1}= a^2_{n}-2
a
n
+
1
=
a
n
2
−
2
for any
n
=
1
,
2
,
.
.
.
n=1,2,...
n
=
1
,
2
,
...
.a) Find
lim
n
→
∞
a
n
+
1
a
1
a
2
.
.
.
a
n
\lim_{n \rightarrow \infty}\frac{a_{n+1}}{a_1a_2 ...a_{n}}
lim
n
→
∞
a
1
a
2
...
a
n
a
n
+
1
b) Find
lim
ν
→
∞
(
1
a
1
+
1
a
1
a
2
+
.
.
.
+
1
a
1
a
2
.
.
.
a
ν
)
\lim_{\nu \rightarrow \infty}\left(\frac{1}{a_1}+\frac{1}{a_1a_2}+...+\frac{1}{a_1a_2 ...a_{\nu}}\right)
lim
ν
→
∞
(
a
1
1
+
a
1
a
2
1
+
...
+
a
1
a
2
...
a
ν
1
)
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