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Infinite Series
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
CHAPTER
8
SEQUENCES OF REAL NUMBERS
INFINITE SERIES
THE INTEGRAL TEST AND COMPARISON TESTS
ALTERNATING SERIES
ABSOLUTE CONVERGENCE AND THE RATIO TEST
POWER SERIES
TAYLOR SERIES
APPLICATIONS OF TAYLOR SERIES
FOURIER SERIES
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8.2
INFINITE SERIES
Partial Sums of Sequences
For any sequence
terms together.
, suppose we start adding the
We define the partial sums S1, S2, . . . , Sn, . . . by
Slide 3
8.2
INFINITE SERIES
Infinite Series
As we add together more and more terms of the sequence
the partial sums draw closer and closer to 1.
In view of this, we write
This new mathematical object,
, is called a series
(or infinite series). It is not a sum in the usual sense of the
word, but rather the limit of the sequence of partial sums.
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8.2
INFINITE SERIES
Infinite Series
For the sequence
, consider the partial sums
Notice that
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8.2
INFINITE SERIES
DEFINITION 2.1
If the sequence of partial sums
converges (to
some number S), then we say that the series
converges (to S) and write
In this case, we call S the sum of the series.
Alternatively, if the sequence of partial sums
diverges (i.e.,
does not exist), then we say that
the series diverges.
Slide 6
8.2
INFINITE SERIES
EXAMPLE 2.2
A Divergent Series
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8.2
INFINITE SERIES
EXAMPLE 2.2
A Divergent Series
Solution The nth partial sum is
and
Since the sequence of partial sums diverges, the series
diverges also.
Slide 8
8.2
INFINITE SERIES
EXAMPLE 2.3
A Series with a Simple Expression for the
Partial Sums
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8.2
INFINITE SERIES
EXAMPLE 2.3
A Series with a Simple Expression for the
Partial Sums
Solution
From the graph, it appears
that the partial sums are
approaching 1, as n → ∞.
Caution: it is extremely
difficult to look at a graph
or a table of any partial sums
and decide whether a given
series converges or diverges.
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8.2
INFINITE SERIES
EXAMPLE 2.3
A Series with a Simple Expression for the
Partial Sums
Solution
(partial fraction
decomposition)
Consider the nth partial sum:
Slide 11
8.2
INFINITE SERIES
EXAMPLE 2.3
A Series with a Simple Expression for the
Partial Sums
Solution
Every term in the partial sum is canceled by another term
in the sum (the next term). For this reason, such a sum is
referred to as a telescoping sum (or collapsing sum).
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8.2
INFINITE SERIES
EXAMPLE 2.3
A Series with a Simple Expression for the
Partial Sums
Solution We now have
The series converges to 1.
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8.2
INFINITE SERIES
THEOREM 2.1
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8.2
INFINITE SERIES
PROOF
Subtracting,
Dividing both sides by (1 – r ,)
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8.2
INFINITE SERIES
EXAMPLE 2.4
A Convergent Geometric Series
Slide 16
8.2
INFINITE SERIES
EXAMPLE 2.4
A Convergent Geometric Series
Solution It appears from the graph that the sequence of
partial sums is converging to some number
around 0.8.
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8.2
INFINITE SERIES
EXAMPLE 2.4
A Convergent Geometric Series
Solution The series is geometric:
Slide 18
8.2
INFINITE SERIES
EXAMPLE 2.4
A Convergent Geometric Series
Solution
Since |r| = 1/3 < 1, we have from Theorem 2.1 that the
series converges to
Slide 19
8.2
INFINITE SERIES
EXAMPLE 2.5
A Divergent Geometric Series
Slide 20
8.2
INFINITE SERIES
EXAMPLE 2.5
A Divergent Geometric Series
Solution
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8.2
INFINITE SERIES
EXAMPLE 2.5
A Divergent Geometric Series
Solution This is a geometric series with ratio r = −7/2.
Since
the series is divergent.
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8.2
INFINITE SERIES
THEOREM 2.2
Slide 23
8.2
INFINITE SERIES
PROOF
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8.2
INFINITE SERIES
kth-Term Test for Divergence
This very useful divergence test follows directly from
Theorem 2.2.
It says that if the terms don’t tend to zero, the series is
divergent and there’s nothing more to do.
However, if the terms do tend to zero, the series may or
may not converge and additional testing is needed.
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8.2
INFINITE SERIES
REMARK 2.1
The converse of Theorem 2.2 is false. That is, having
does not guarantee that the series
converges.
Be very clear about this point. This is a very common
misconception.
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8.2
INFINITE SERIES
EXAMPLE 2.6
A Series Whose Terms Do Not Tend to
Zero
Slide 27
8.2
INFINITE SERIES
EXAMPLE 2.6
A Series Whose Terms Do Not Tend to
Zero
Solution
The partial sums appear to be
increasing without bound as n
increases.
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8.2
INFINITE SERIES
EXAMPLE 2.6
A Series Whose Terms Do Not Tend to
Zero
Solution Note that
So, by the kth-term test for divergence, the series must
diverge.
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8.2
INFINITE SERIES
EXAMPLE 2.7
The Harmonic Series
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8.2
INFINITE SERIES
EXAMPLE 2.7
Solution
The Harmonic Series
The series might converge to
a number around 3.6.
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8.2
INFINITE SERIES
EXAMPLE 2.7
The Harmonic Series
Solution Note that
Be careful: this does not say that the series converges.
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8.2
INFINITE SERIES
EXAMPLE 2.7
The Harmonic Series
Consider the nth partial sum
Solution
Sn corresponds to the sum of
the areas of the n rectangles
superimposed on the graph of
y = 1/x.
Slide 33
8.2
INFINITE SERIES
The Harmonic Series
EXAMPLE 2.7
Since each of the indicated rectangles lies partly
Solution above the curve,
The sequence
since
diverges,
Slide 34
8.2
INFINITE SERIES
EXAMPLE 2.7
The Harmonic Series
Solution
Slide 35
8.2
INFINITE SERIES
THEOREM 2.3
The proof of the theorem is left as an exercise.
Slide 36