Calculus 2 : Harmonic Series
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Example Questions
Example Question #1 : Harmonic Series
The Harmonic series is a special case of a 
A 
Example Question #2 : Harmonic Series
Which of the following tests will help determine whether 
Integral Test: The improper integral determines that the harmonic series diverge.
P-Series Test: The summation converges since 
Root Test: Since the limit as 
Integral Test: The improper integral determines that the harmonic series diverge.
The series 
The Nth term test and the Divergent test may not be used to determine whether this series converges, since this is a special case. The root test also does not apply in this scenario.
According the the P-series Test, 
This leaves us with the Integral Test.
Since the improper integral diverges, so does the series.
Example Question #2 : Series Of Constants
Determine whether the following series converges or diverges:
The series (absolutely) converges
The series conditionally converges
The series diverges
The series may (absolutely) converge, diverge, or conditionally converge
The series (absolutely) converges
Given just the harmonic series, we would state that the series diverges. However, we are given the alternating harmonic series. To determine whether this series will converge or diverge, we must use the Alternating Series test.
The test states that for a given series 
First, we must evaluate the limit of 
The limit equals zero because the numerator of the fraction equals zero as n approaches infinity.
Next, we must determine if 
Because both parts of the test passed, the series is (absolutely) convergent.
Example Question #1 : Harmonic Series
Consider the alternating series
Which of the following tests for convergence is NOT conclusive?
The root test
The alternating series test
The ratio test
The limit test for divergence
The limit test for divergence
Let
be the nth summand in the series. The limit test for divergence states that
implies that the series diverges.
However,
so the test is inconclusive.
Example Question #5 : Harmonic Series
Does the following series converge?
Cannot be determined
No
Yes
No
No the series does not converge. The given problem is the harmonic series, which diverges to infinity.
Example Question #6 : Harmonic Series
Does the following series converge?
No
Yes
Cannot be determined
Yes
The series converges. The given problem is the alternating harmonic series, which converges by the alternating series test.
Example Question #7 : Harmonic Series
Which of the following tests can be used to (successfully) test for the convergence/divergence of the harmonic series?
The Limit Test for Divergence
The Root Test
None of the given tests can be used.
The Ratio Test
The Integral Test
The Integral Test
Only the Integral Test will work on the Harmonic Series, 
To use the Integral Test, we evaluate
Since 
The Ratio Test and the Root Test will always yield the same conclusion, so if one test fails, the both fail and vise versa.
For the Ratio Test,
Example Question #8 : Harmonic Series
Let's say you are given harmonic series in the following form:
You are then asked to determine if the series converges, or diverges. For what values of p would this series be convergent? Assume p>0.
The given series is called generalized harmonic series.
The series converges, if 
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