section 8.4: other convergence tests practice hw from stewart textbook (not to hand in) p. 592 #...
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Section 8.4: Other Convergence Tests
Practice HW from Stewart Textbook (not to hand in)
p. 592 # 3-7, 12-17, 19-25 odd, 33
Alternating Series
An alternating series is a series whose terms
alternate (change) in sign (from + to -)
Examples: and .
1
1
12
)1(
n
n
n
n
12 1
)1(
n
n
n
n
Alternating Series Test
Let . The alternating series
and
converge if the following two conditions are met.
0nb
1
1)1(n
nn b
1
)1(n
nnb
Note: The alternating series test is only a test for
convergence. If it fails, the series may still
converge or diverge.
Estimating the Sum of a Convergent Alternating Series
Most of the series test (geometric series is one
exception) so far have been designed to only
indicate whether a given series converges or
diverges. When the series is convergent, no
method for find the sum (the
value the series converges to) has been given.
However, it is possible to estimate the sum of the
series to a designated accuracy. Here, we
present a method for doing this that is designed
for alternating series (note others exist for other
types of series – for example, see p. 583 for a
method involving the integral test).
Error Estimate for a Convergent Alternating Series
Let be the sum of a convergent
alternating series. If = the sum of the first
n – 1 terms of the series and let = the
difference between s and . Then
1)1( nnbs
1ns
1nR
1ns
nnn bssR || || |error| sum 1)-(n partial
and sumbetween Error 1|1th
Note
1.Approximating the sum up to a certain number of decimal places means the digits of approximation and the sum agree up to amount of decimal places. In general, for error estimation, this means that to agree up to n decimal places,
9 000||zeros
n
error
2.Once the number of terms needed in the series to approximate the sum to the desired accuracy is found, Maple can be useful for getting the approximation.
Example 4: Approximate the sum of the following
series to 3 decimal places.
Solution: (In typewritten notes)
25
1
16
1
9
1
4
11
)1(
12
1
n
n
n
Absolute Convergence
Definition: A series is called absolutely
convergent if the series is convergent.
Fact: If a series is absolutely convergent, then
the series is itself convergent. Sometimes, it can
be easier to determine that a series is absolutely
convergent.
na
na
Example 7: Demonstrate that the series ,
while not absolutely convergent, is still
convergent.
Solution:
1
1
1
)1(
n
n
n
A good test sometimes for determining absolute
convergence (and or convergence) of a series is
the ratio test, which we state next.
Ratio Test
Given an infinite series .
1. converges absolutely (and therefore converges) if .
2. diverges if or .
3. The ratio test is inconclusive if .
na
na
1 lim 1
n
n
n a
a
na 1 lim 1
n
n
n a
a
n
n
n a
a 1 lim
1 lim 1
n
n
n a
a
Example 8: Determine if the series
is absolutely convergent. In addition, determine if
the series is convergent or divergent.
Solution:
12
)2(
n
n
n
Note: The ratio test does not apply just to
alternating series. It can be used to analyze the
absolute convergence and overall convergence of
any series.
Example 9: Determine if the series
is absolutely convergent. In addition, determine if
the series is convergent or divergent.
Solution:
1 !
3
n
n
n
Note: The ratio test is inconclusive if .
In this case, other series techniques to analyze
the convergence and divergence of a series must
be used.
1lim 1
n
n
n a
a
Note: The ratio test is inconclusive if .
In this case, other series techniques to analyze
the convergence and divergence of a series must
be used.
1lim 1
n
n
n a
a