# AP Calculus BC : Series of Constants

## Example Questions

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### Example Question #1 : Series In Calculus

Consider:  .   Will the series converge or diverge? If converges, where does this coverge to?

Explanation:

This is a geometric series.  Use the following formula, where  is the first term of the series, and  is the ratio that must be less than 1.  If  is greater than 1, the series diverges.

Rationalize the denominator.

### Example Question #1 : Geometric Series

Consider the following summation:  .  Does this converge or diverge?  If it converges, where does it approach?

Explanation:

The problem can be reconverted using a summation symbol, and it can be seen that this is geometric.

Since the ratio is less than 1, this series will converge.  The formula for geometric series is:

where  is the first term, and  is the common ratio.  Substitute these values and solve.

### Example Question #3 : Series In Calculus

A worm crawls up a wall during the day and slides down slowly during the night. The first day the worm crawls one meter up the wall. The first night the worm slides down a third of a meter. The second day the worm regains one third of the lost progress and slides down one third of that distance regained on the second night. This pattern of motion continues...

Which of the following is a geometric sum representing the distance the worm has travelled after  12-hour periods of motion? (Assuming day and night are both 12 hour periods).

Explanation:

The sum must be alternating, and after one period you should have the worm at 1m. After two periods, the worm should be at 2/3m. There is only one sum for which that is true.

### Example Question #7 : Concepts Of Convergence And Divergence

Determine whether the following series converges or diverges. If it converges, what does it converge to?

Explanation:

First, we reduce the series into a simpler form.

We know this series converges because

By the Geometric Series Theorem, the sum of this series is given by

### Example Question #1 : Geometric Series

Calculate the sum of a geometric series with the following values:,,. Round the answer to the nearest integer.

Explanation:

This is a geometric series.

The sum of a geometric series can be calculated with the following formula,

, where n is the number of terms to sum up, r is the common ratio, and  is the value of the first term.

For this question, we are given all of the information we need.

Solution:

Rounding,

### Example Question #2 : Geometric Series

Calculate the sum, rounded to the nearest integer, of the first 16 terms of the following geometric series:

Explanation:

This is a geometric series.

The sum of a geometric series can be calculated with the following formula,

, where n is the number of terms to sum up, r is the common ratio, and  is the value of the first term.

We have  and n and we just need to find r before calculating the sum.

Solution:

### Example Question #3 : Geometric Series

Calculate the sum of a geometric series with the following values:

,, ,

rounded to the nearest integer.

Explanation:

This is a geometric series.

The sum of a geometric series can be calculated with the following formula,

, where n is the number of terms to sum up, r is the common ratio, and  is the value of the first term.

For this question, we are given all of the information we need.

Solution:

Rounding,

### Example Question #1 : Alternating Series With Error Bound

Determine whether the following series converges or diverges:

The series may (absolutely) converge, diverge, or conditionally converge

The series conditionally converges

The series diverges

The series (absolutely) converges

The series (absolutely) converges

Explanation:

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 where  or  where  for all n, if  and  is a decreasing sequence, then  is convergent.

First, we must evaluate the limit of  as n approaches infinity:

The limit equals zero because the numerator of the fraction equals zero as n approaches infinity.

Next, we must determine if  is a decreasing sequence. , thus the sequence is decreasing.

Because both parts of the test passed, the series is (absolutely) convergent.

### Example Question #2 : Alternating Series With Error Bound

Determine whether

converges or diverges, and explain why.

Convergent, by the alternating series test.

Divergent, by the test for divergence.

Convergent, by the -series test.

More tests are needed.

Divergent, by the comparison test.

Convergent, by the alternating series test.

Explanation:

We can use the alternating series test to show that

converges.

We must have   for  in order to use this test. This is easy to see because  is in for all  (the values of this sequence are ), and sine is always nonzero whenever sine's argument is in .

Now we must show that

1.

2.  is a decreasing sequence.

The limit

implies that

so the first condition is satisfied.

We can show that  is decreasing by taking its derivative and showing that it is less than  for :

The derivative is less than , because  is always less than , and that  is positive for , using a similar argument we used to prove that  for . Since the derivative is less than  is a decreasing sequence. Now we have shown that the two conditions are satisfied, so we have proven that

converges, by the alternating series test.

### Example Question #3 : Alternating Series With Error Bound

For the series:  , determine if the series converge or diverge.  If it diverges, choose the best reason.

Explanation:

The series given is an alternating series.

Write the three rules that are used to satisfy convergence in an alternating series test.

For :

The first and second conditions are satisfied since the terms are positive and are decreasing after each term.

However, the third condition is not valid since  and instead approaches infinity.