(a)
Show whether the infinite series converges.
(a)
Answer to Problem 54E
The infinite series diverges by the Limit Comparison Test.
Explanation of Solution
Given information:
To use with Limit Comparison Test, find a series
Thus,
Then
Apply the limit comparison test:
Since the limit is a constant,
But
We have
Series
Then
Therefore,
By Limit Comparison Test, the infinite series diverges.
(b)
Explain whether the infinite series converges.
(b)
Answer to Problem 54E
The infinite series diverges.
Explanation of Solution
Given information:
We have
Use nth − term test:
Since k approaches infinity, we will take the limit of the original series.
Then simplify:
The series does not approach zero.
Therefore,
The infinite series diverges.
(c)
Discuss whether the infinite series converges.
(c)
Answer to Problem 54E
The infinite series absolutely converges by the Direct Comparison Test.
Explanation of Solution
Given information:
To use with Limit Comparison Test, find a series
Thus,
Since
Consider the absolute value of the terms:
Now,
Perform the Direct Comparison Test:
Compare the numerators and denominators.
Smaller numerator divided by larger denominators equals the smaller number.
Since
Then
According to the definition of Absolute Convergence,
(d)
Decide whether the infinite series converges.
(d)
Answer to Problem 54E
The infinite series diverges by the Integral Test.
Explanation of Solution
Given information:
We have
Use the Integral Test:
Take the anti-derivative:
Solve the integral:
The integral comes to infinity.
Therefore,
The series diverges by the Integral Test.
Chapter 10 Solutions
Calculus: Graphical, Numerical, Algebraic
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