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Series |
Absolute
convergence |
Conditional
convergence |
Series
of positive terms |
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Absolute
convergence of a series |
An
infinite series |
![](InfSer.GIF) |
is called absolutely convergent if the infinite series |
![](InfSerA.GIF) |
is convergent, i.e., |
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the series
of absolute values of its
terms converge. |
Thus,
if a series is absolutely convergent, then both series, |
![](InfSerB.GIF) |
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Example: Let
show that the series |
![](SerAltEx2.gif) |
is
absolutely convergent. |
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Solution: Given
series is absolutely
convergent since the series
of absolute values of its
terms |
|
![](SerAltEx2A.gif) |
converges
to 2, |
![](SerAltEx2B.gif) |
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Therefore,
the series |
![](SerAltEx2C.gif) |
is
convergent and its sum is 2/3. |
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Example: Let
show that the series |
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is
absolutely convergent. |
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Solution: Given
series is absolutely
convergent since the series
of absolute values of its
terms |
![](SerAltEx3A.gif) |
Therefore,
the series converges and its sum |
![](SerAltEx3B.gif) |
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Conditional
convergence |
If
an infinite series |
![](InfSer.GIF) |
is not convergent, it is called divergent. However, if it is convergent
but not |
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absolutely convergent,
it is called conditionally convergent. |
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Series
of positive terms |
Suppose
the series |
![](SerA.gif) |
consists
of positive terms only. Then its partial sums |
![](SerPartS.gif) |
form
an |
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increasing sequence, therefore given series converges if and only if
the sequence of partial sums {sn}
is |
bounded. |
We use this property for convergence tests of series with positive
terms. |
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Example: The series of positive terms |
![](SerPosTEx.gif) |
converges
since |
![](SerPosTExA.gif) |
that
is, the sequence of partial sums is
bounded above, and the series converges to the number e. |
If
we calculate the partial sum of first 10 terms of the series,
s10
= 2.718281
obtained
is the number e
to six
correct
decimal digits
what shows that the series
increases relatively fast. |
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