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Series |
Infinite series |
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Convergence
of infinite series |
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Divergence
of infinite series |
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Convergent
and divergent series
examples |
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Harmonic
series |
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| Convergence
of infinite series |
| If
the
limit of the sequence of partial sums
exists
as a real number, then the series is convergent. |
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| Divergence
of infinite series |
| If
the limit of the sequence of partial sums does not exist or tends to
+
oo
or -
oo, |
 |
| then,
the series is divergent or oscillates. |
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| Convergent
and divergent series
examples |
| Example: Show
that the series |
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diverges. |
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| Solution:
The given
infinite sum of natural numbers is called the arithmetic series. |
| Using
the formula for the sum
of the arithmetic sequence, whose difference d
= 1,
we calculate the sum of |
| the first n
terms of the series. |
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| Example: Let
show
that the series |
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converges. |
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| Solution:
We
use the method of partial
fraction decomposition to rearrange given expression. |
| Since
the denominator consists of linear factors then, we can write |
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| Therefore, |
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| Since
the second and the first term of any two successive parentheses cancel
then,
the sum of
the first n
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terms of the series |
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therefore the series converge, and its sum s
= 1. |
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| Example: Show
that the series |
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converges. |
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| Solution:
Use the method of partial
fraction decomposition to rearrange given expression. |
| Since
the
number of the coefficients used in the expansion relates to the degree
of the polynomial in the |
| denominator,
and as it consists of linear factors some of which are repeated
then, |
 |
| Therefore, |
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| Since
the second and the first term of any two successive parentheses cancel
then, the sum
of
the first n
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terms of the series |
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therefore the series converge, and its sum s
= 1. |
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| Harmonic
series |
| The
series of the reciprocals of natural numbers |
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| is
called the harmonic series since the middle term of any three successive
terms is the harmonic mean of |
| the
other two. |
| Given
three positive numbers, a,
b
and c
are said to be in the harmonic proportion if |
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| that
is, if b
is the harmonic mean of the numbers, a
and c. |
| Therefore,
any three successive terms, |
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of
the harmonic series, are in the harmonic |
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| proportion since |
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| As,
in the harmonic proportion a
> c then, a
-
b > b -
c therefore a
+ b > 3b -
c |
| or
a
+ b + c > 3b. |
| That
is, the sum of any three successive terms of the harmonic series is
three times greater than the middle |
| term.
P. Mengoli (1626 - 1686) used this property to prove divergence of the
harmonic series. |
| Let
examine behavior of the sequence of partial sums of the harmonic series
applying this property. |
| Thus,
the sum of the three successive terms beginning from the second gives |
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| Next
nine terms, from n = 5 to
13, divide in three groups with three terms in
each, so that; first gives |
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| Therefore, |
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| since
the parentheses
equals already known value s3. |
| For
next 27
terms, from n =
14 to 40,
following the same procedure, we get that their sum is greater
than |
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for
which is already shown that is >
1,
thus s3 +
9 + 27
> 3. |
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| Generally,
it can be written |
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that
is, the partial sums sn increase
to infinity as n
®
oo. |
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| Therefore,
the harmonic series diverges. |
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