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SEEMOUS
2014 SEEMOUS
Problem 2
convergence of recurrence
convergence of recurrence
Source: SEEMOUS 2014 P2
June 4, 2021
Sequences
limits
Problem Statement
Consider the sequence
(
x
n
)
(x_n)
(
x
n
)
given by
x
1
=
2
,
x
n
+
1
=
x
n
+
1
+
x
n
2
+
2
x
n
+
5
2
,
n
≥
2.
x_1=2,\enspace x_{n+1}=\frac{x_n+1+\sqrt{x_n^2+2x_n+5}}2,\enspace n\ge2.
x
1
=
2
,
x
n
+
1
=
2
x
n
+
1
+
x
n
2
+
2
x
n
+
5
,
n
≥
2.
Prove that the sequence
y
n
=
∑
k
=
1
n
1
x
k
2
−
1
,
n
≥
1
y_n=\sum_{k=1}^n\frac1{x_k^2-1},\enspace n\ge1
y
n
=
∑
k
=
1
n
x
k
2
−
1
1
,
n
≥
1
is convergent and find its limit.
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