a.
To find: The interval of convergence of the series.
a.
Answer to Problem 49E
The interval of convergence of the series is
Explanation of Solution
Given information:
Calculation:
By applying the Ratio Test and looking at the interval's endpoints, one can determine the interval where the series will converge.
Taking the absolute value of the terms in the series and looking at the following limit, only using the Ratio Test for series with non-negative terms.
The series converges absolutely based on the previous step if
The left endpoint of the interval is given by,
Where,
The series does not absolutely converge at the left endpoint since the
Analyze if the series conditionally converges at the left endpoint. The series at the left endpoint according to the preceding step is
The series converges at the left endpoint according to the Alternating Series Test because
The right endpoint of the interval is given by,
Where,
The series does not converge at the proper endpoint since it has only positive terms, is absolute convergent if it does, never converges conditionally, and diverges according to the
From the above steps it is known that,
The interval of convergence of the series is,
Therefore, the interval of convergence of the series is
b.
To find: For what value of
b.
Answer to Problem 49E
The value of
Explanation of Solution
Given information:
Calculation:
From part (a) it is known that,
The interval on which the series converges absolutely is given by,
Using the Ratio test,
The series diverges if,
Then, the series converges absolutely on
Therefore, the value of
c.
To find: For what value of
c.
Answer to Problem 49E
The value of
Explanation of Solution
Given information:
Calculation:
Identifying the intervals in parts (a) and (b) on which the series converges and on which it converges absolutely.
These findings can be used to determine where the series conditionally converges by looking at the conditional convergence of the endpoints.
From part (a) it is known that,
The interval on which the series converges is given by,
But from part (a) and part (b) it is found that it does not converge absolutely at this point.
Then, the series converges conditionally on
Therefore, the value of
Chapter 9 Solutions
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