Calculus 2 : Comparing Series

Example Questions

Example Question #11 : Comparing Series

We consider the following series:

Determine the nature of the convergence of the series.

The series is divergent.

The series is divergent.

Explanation:

We will use the Comparison Test to prove this result. We must note the following:

is positive.

We have all natural numbers n:

, this implies that

.

Inverting we get :

Summing from 1 to , we have

We know that the is divergent. Therefore by the Comparison Test:

is divergent.

Example Question #12 : Comparing Series

Is the series

convergent or divergent, and why?

Divergent, by the comparison test.

Divergent, by the test for divergence.

Convergent, by the ratio test.

Divergent, by the ratio test.

Convergent, by the comparison test.

Convergent, by the comparison test.

Explanation:

We will use the comparison test to prove that

converges (Note: we cannot use the ratio test, because then the ratio will be , which means the test is inconclusive).

We will compare  to  because they "behave" somewhat similarly. Both series are nonzero for all , so one of the conditions is satisfied.

The series

converges, so we must show that

for .

This is easy to show because

since the denominator  is greater than or equal to  for all .

Thus, since

and because

converges, it follows that

converges, by comparison test.

Example Question #11 : Series Of Constants

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

Conditionally converges.

Neither converges nor diverges.

Diverges

Converges

There is not enough information to decide convergence.

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 #1 : Numerical Approximation

For which values of p is

convergent?

it doesn't converge for any values of

only

All positive values of

only

Explanation:

We can solve this problem quite simply with the integral test. We know that if

converges, then our series converges.

We can rewrite the integral as

and then use our formula for the antiderivative of power functions to get that the integral equals

.

We know that this only goes to zero if . Subtracting p from both sides, we get

.

Example Question #14 : Comparing Series

Determine the convergence of the series using the Comparison Test.

Cannot be determined

Series diverges

Series converges

Series converges

Explanation:

We compare this series to the series

Because

for

it follows that

for

This implies

Because the series on the right has an exponent ,

the series on the right converges

making

converge as well.

Example Question #2951 : Calculus Ii

Determine the convergence of the series using the Comparison Test.

Series diverges

Cannot be determined

Series converges

Series diverges

Explanation:

We compare this series to the series

Because

for

it follows that

for

This implies

Because the series on the right has a degree of  equal to  in the denominator,

the series on the right diverges

making

diverge as well.

Example Question #2952 : Calculus Ii

Does the series converge or diverge? If it does converge, then what value does it converge to?

Diverges

Converges to

Converges to

Converges to 1

Converges to

Converges to 1

Explanation:

To show this series converges, we use direct comparison with

,

which converges by the p-series test with .

Thus we must show that

.

Cross multiplying the previous section and multiplying by , we obtain .

Since this holds for all we can conclude that

.

Summing from  to , and noting that

for all , we obtain the following inequality:

.

Therefore the series

converges by direct comparison.

Now to find the value, we note that

,

so that

.

Now let

be a sequence of partial sums.

Then we have

Therefore

.

Taking the limit as , we obtain the following:

Therefore we have

.

Example Question #2953 : Calculus Ii

Does the series converge?

Cannot be determined

Yes

No

No

Explanation:

Notice that for

This implies that

for

Which then  implies

Since the right-hand side is the harmonic series, we have

and thus the series does NOT converge.

Example Question #2954 : Calculus Ii

Determine whether the series converges, absolutely, conditionally or in an interval.

Converges conditionally

Does not converge at all

Converges absolutely

Converges in an interval

Converges absolutely

Explanation:

Example Question #2955 : Calculus Ii

Determine whether the series converges

Converges conditionally

Does not converge at all

Converges absolutely

Converges in an interval