9.2 Series and Convergence

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9.2 Series and Convergence
Riverfront Park, Spokane, WA
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Start with a square one unit by one unit
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This is an example of an infinite series.
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This series converges (approaches a limiting
value.)
Many series do not converge
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The infinite series for a sequence is written as
. . .
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In an infinite series
a1, a2, are terms of the series.
an is the nth term.
Partial sums
nth partial sum
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Ex. 1 Find the nth partial sum of the series
Whats the pattern for each sum??
The denoms are all and the numerators??
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Then what is the infinite sum of the series??
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So the series converges to 1.
This is called a geometric series.
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Geometric Series
In a geometric series, each term is found by
multiplying the preceding term by the same
number, r, called the common ratio.
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Example 2
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Example 3
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The nth partial sum of a geometric series is
This is a handy formula to know, as you may be
asked to find an nth partial sum of a geometric
series!
From this, you can see where formula for the
infinite sum comes from
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Ex. 4 Find the partial sum of the series
Write out a few terms
Since the middle terms zero out, we are left
with
The series converges to 1.
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The last problem is an example of a telescoping
series
Since the middle terms will always cancel out, we
can find the partial sum of a telescoping series
by
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Ex. 5 Find the sum of the series
This can be written in telescoping form using
partial fractions
So the nth partial sum is
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Ex. 6 a) Find the sum (if it exists) of the
series
Re-write
Since we know that the series
converges to the sum of
b) Find the sum (if it exists) of the series
Since the series diverges.
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Ex. 7 Use a geometric series to express
as the ratio of two integers. (as a
rational number)
Note that we can write as
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nth Term Test for Divergence
The first requirement for convergence of a series
is that the terms of the sequence must approach
zero. That is,
If converges, then
The contrapositive of this fact leads us to a
useful test for divergence
If , then
diverges.
(Notice we are talking about the limit of the
sequence, not the partial sum!)
So how could we use this test? Lets take a look
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Ex. 8
Determine if the series diverges
a)
Lets look at the sequence
Since the limit of the sequence is not 0, by the
nth term test, the series diverges!
b)
Since the limit of the sequence is not 0, by the
nth term test, the series diverges!
The limit of the nth term is 0, so the test
fails! We cant draw a conclusion about
convergence or divergence.
c)
0
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Bouncing Ball Problem
A ball is dropped from a height of 6 feet and
begins bouncing. The height of each bounce is
three-fourths the height of its previous bounce.
Find the total vertical distance traveled by the
ball.
Let D1 be the initial distance the ball travels
Let D2 be the distance traveled up and down
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Notice this is a geometric series!
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