§2. Convergent fractions
If to count some makers from the beginning of the continued fraction, to separate them out and to convert their meaning into a common fraction, consequently the appeared fraction is called a convergent fraction.
Theorem3.
In
order to find the segment at the place of
it is necessary to multiply
the segment of the previous convergent n
by the
appropriate multiple
ға),
to
add the segment of the fraction at the place
to
the multiplied one.
This way it is necessary to multiply the numerator at the place
by the appropriate multiple
and
to add the numerator of the fraction
to the multiplied one.
Proof. Let’s take the following continued fraction:
(2.1)
Let’s have its convergents like this:
(2.2)
the equations there would be the following:
(2.3)
(2.4),
Let’s
say that the previous rule influences the making of the convergent
.
So
we will have this expression:
(2.5)
Now if we show that the following equation is appropriate, the theorem is solved
(2.6).
It is clear that the meaning of the convergent is actually this:
(2.7)
and
the meaning of the fraction
is:
.
If
we put expression
instead
of the
of the convergent No
,
we
will see the convergent
will
have equal value.
If it is so, let’s take the equation (2.6)
(2.7)
and open brackets of the last expression (2.7)
(2.8)
If to multiple both members of the right denominator of the expression (2.8) by
(2.9)
we will have the above expression. At the end if to recall the condition of the equation (2.5), the expression (2.9) will be as the following
(2.10).
The last equation is equal to the equation (2.6). That is why we are convinced that mentioned condition influence the convergent , i.e. the theorem is proved.
We have known that this line influences the third convergent. Hence according to the last conclusion that line influences the fourth convergent. If it influences the fourth fraction, it also influences the fifth one and so on.
In
the above continued fraction the first convergent must be 3, the
second convergent must be
the
third convergent
must be
the
fourth convergent
must be
the
fifth convergent
must be
.
The
sixth convergent is equal to the value of the given continued
fraction.
It is easy to count the convergent fractions in the following way:
-
2
4
5
3
10
28
102
533
1
3
7
31
162
Here
the top numbers are the multiples of the given continued fraction,
the lower lines show the segments and numerators of the convergent
fractions.
Hereof we can defind the first two convergents as 3 and
,
other
convergents are found according to the previous found approach.
Theorem 4. The value of the finite continued fraction is between values of the neighbouring convergents; and it is closer to the value of the behind convergent rather that to the value of the following one.
Proof.
Let’s take a
finite fraction and a continued chain
)
with
the value y.
So
(2.11)
the above condition is implied. And now let’s take convergents and three sequent makers. For example, they have the following meaning:
(2.12)
According to the previously defined conclusion this equation is appropriate:
(2.13)
If
we put the expression
(2.14)
instead of a
in
the right denominator
of the equation (2.13), so it will be an actual value of the continued fraction. Hereof the following equation will be done
(2.15).
According to the equation (2.15) we will have the following equation:
(2.16)
According
to our project as the numbers
and
are positiv,
so the differences inside the brackets of the equation
(2.16)
should be equally positive or equally negative.
If
both differences are positive:
they
will have this expression
(2.17)
If both differences are negative:
they
will have this expression
(2.18)
It
is shown that the meaning of
in the equations (2.17) and (2.18) is between two sequent fractions
and
.
Also a condition
of
the equation (2.16)
is applied and as it is
,
the meaning of the difference of
is
less that the difference of
(
)
. Hereof
we
will have this equation:
(2.19)
The
equation (2.19) shows that the value of
-
is closer to the fraction
rather
than to the value of
-.
Hence
the theorem is proved.
Note:
if
the actual value of the fraction
)
is
,
so
it is clear that the number is greater than the number
or
.
When with the equations (2.17) and (2.18) it is
,
the condition is equation
.
So
finite equations
,
,
...
are performed.
Hereof we see that the actual value of the continued fraction is greater than convergents at every even place and is less that the convergents at every uneven place.