# AP Calculus BC : P-Series

## Example Questions

### Example Question #1 : P Series

One of the following infinite series CONVERGES. Which is it?

Possible Answers:

None of the others converge.

Correct answer:

Explanation:

converges due to the comparison test.

We start with the equation . Since  for all values of k, we can multiply both side of the equation by the inequality and get  for all values of k. Since  is a convergent p-series with   hence also converges by the comparison test.

### Example Question #4 : Comparing Series

Determine the nature of convergence of the series having the general term:

Possible Answers:

The series is convergent.

The series is divergent.

Correct answer:

The series is convergent.

Explanation:

We will use the Limit Comparison Test to establish this result.

We need to note that the following limit

goes to 1 as n goes to infinity.

Therefore the series have the same nature. They either converge or diverge at the same time.

We will focus on the series:

.

We know that this series is convergent because it is a p-series. (Remember that

converges if p>1 and we have p=3/2 which is greater that one in this case)

By the Limit Comparison Test, we deduce that the series is convergent, and that is what we needed to show.

### Example Question #2 : P Series

Determine if the series converges or diverges. You do not need to find the sum.

Possible Answers:

Diverges

Neither converges nor diverges.

Conditionally converges.

There is not enough information to decide convergence.

Converges

Correct answer:

Converges

Explanation:

We can compare this to the series  which we know converges by the p-series test.

To figure this out, let's first compare  to . For any number n,  will be larger than .

There is a rule in math that if you take the reciprocal of each term in an inequality, you are allowed to flip the signs.

Thus,  turns into

.

And so, because  converges, thus our series also converges.

### Example Question #3 : P Series

Which of the following tests will help determine whether   is convergent or divergent, and why?

Possible Answers:

Divergence Test: Since limit of the series approaches zero, the series must converge.

Nth Term Test: The series diverge because the limit as  goes to infinity is zero.

Integral Test: The improper integral determines that the harmonic series diverge.

P-Series Test: The summation converges since .

Root Test: Since the limit as  approaches to infinity is zero, the series is convergent.

Correct answer:

Integral Test: The improper integral determines that the harmonic series diverge.

Explanation:

The series  is a harmonic 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,  must converge only if .  Therefore this could be a valid test, but a wrong definition as the answer choice since the series diverge for .

This leaves us with the Integral Test.

Since the improper integral diverges, so does the series.

### Example Question #4 : P Series

Does the series  converge conditionally, absolutely, or diverge?

Possible Answers:

Diverges.

Converge Absolutely.

Does not exist.

Cannot tell with the given information.

Converge Conditionally.

Correct answer:

Converge Conditionally.

Explanation:

The series converges conditionally.

The absolute values of the series  is a divergent p-series with .

However, the the limit of the sequence  and it is a decreasing sequence.

Therefore, by the alternating series test, the series converges conditionally.

### Example Question #5 : P Series

True or False, a -series cannot be tested conclusively using the ratio test.

Possible Answers:

False

True

Correct answer:

True

Explanation:

We cannot test for convergence of a -series using the ratio test. Observe,

For the series ,

.

Since this limit is  regardless of the value for , the ratio test is inconclusive.