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Series |
Infinite series |
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The sequence of partial
sums |
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The
sum of the 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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| Infinite series |
| An
infinite series is the sum of infinite sequence of terms which we
denote |
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| That
is, given an infinite sequence of real numbers, a1,
a2, a3, . . . , an,
. . .
all of the terms of which are added
together, where an
denotes the general term of the series. |
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| The
sequence of partial
sums |
| Recall
that we've already dealt with infinite series in previous section when
representing real numbers as infinite
decimals. Thus, for example the real number |
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| so,
when adding up this infinite number of terms, we start by calculating
its partial sums |
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| Thus,
obtained is sequence {sn}
of the partial sums, s1,
s2, s3, . . . , sn,
. . .
where sn
denotes the sum of
the first n
terms of the series and is called the nth
partial sum. |
| Therefore,
the
limit of the sequence {sn}
as n
tends to infinity, i.e., |
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equals the value of the real number 0.333
. . .
that is, equals the sum of
the infinite series. |
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The
sum of the series |
| An infinite series has a sum if the sequence of its partial sums
converge to a finite number s,
i.e., |
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| The sum of
an infinite series is the limit of its partial sums
as n
tends to infinity. |
| Let
rewrite the above example using this notation, |
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| where, |
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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, |
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| 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
terms of the series |
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therefore the series converge, and its sum s
= 1. |
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