CONTINUED FRACTIONS
ROMAN VERSHYNIN, MATH 451, FALL 2011
As an application of the theory of limits, we will verify that
(1)
1 +
1
2 +
1
2 +
1
2 +
···
=
√
2
.
An object in the left side of the equation is called a
continued fraction
. Equation (1)
is one example of a rich theory of continuous fractions, see Wikipedia if interested.
One can rigorously deﬁne the continuous fraction in (1) as a limit of the sequence
of ﬁnite fractions of the form
1
,
1 +
1
2
,
1 +
1
2 +
1
2
,
1 +
1
2 +
1
2 +
1
2
,...
The ﬁrst six terms of this sequence are approximately
1
,
1
.
5
,
1
.
4
,
1
.
417
,
1
.
4138
,
1
.
41429
,
so the sequence indeed seems to converge to
√
2
≈
1
.
414214 quite fast.
The following theorem is a rigorous way to state this convergence.
Theorem 1.
Let
a
1
= 1
and
(2)
a
n
+1
= 1 +
1
1 +
a
n
,
n
= 1
,
2
,....
Then
lim
a
n
=
√
2
.
We shall prove the theorem by the following series of results.
Lemma 2.
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 Fall '08
 STAFF
 Calculus, Fractions, Limits, lim, Mathematical analysis, weierstrass theorem, Continued fraction

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