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Convergence of a Recursive Sequence

​
n
1
The limit of the sequence
1,1+
1
1
,1+
1
1+
1
1
,1+
1
1+
1
1+
1
1
,…
, or equivalently
a
1
=1,
a
n+1
=1+
1
a
n
, satisfies the equation
x=1+
1
x
. The solutions are
1+
5
2
,
1-
5
2
. In the graphic we show that the limit is the golden ratio
1+
5
2
.
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