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AP Calculus BC : Series of Constants

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Example Question #13 : Ratio Test And Comparing Series

Determine what the following series converges to using the Ratio Test, and whether the series is convergent, divergent or neither.

$\sum_{n=0}^{\infty} \frac{(n+1)!(-5)^n}{n!\2^{n+1}}$

Possible Answers:

$\infty$ , and Divergent

$\frac{5}{2}$ , and Neither

$\frac{5}{2}$ , and Convergent

$\frac{5}{2}$ , and Divergent

$\infty$ , and Convergent

Correct answer:

$\frac{5}{2}$ , and Divergent

Explanation:

To determine if

$\sum_{n=0}^{\infty} \frac{(n+1)!(-5)^n}{n!\2^{n+1}}$

is convergent, divergent or neither, we need to use the ratio test.

The ratio test is as follows.

Suppose we a series $\sum a_n$ . Then we define,

$L=\lim_{n\rightarrow \infty}\left | \frac{a_{n+1}}{a_n}\right |$ .

L<1 the series is absolutely convergent (and hence convergent).

L>1 the series is divergent.

L=1 the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply this to our situtation.

Let

$a_n=\frac{(n+1)!(-5)^n}{n!\2^{n+1}}$

and

$a_{n+1}=\frac{(n+2)!(-5)^{n+1}}{(n+1)!\2^{n+2}}$

Now

$L=\lim_{n\rightarrow \infty}\left | \frac{\frac{(n+2)!(-5)^{n+1}}{(n+1)!\2^{n+2}}}{\frac{(n+1)!(-5)^n}{n!\2^{n+1}}}\right |$

We can rearrange the expression to be

$L=\lim_{n\rightarrow \infty}\left | \frac{n!(n+2)!\(-5)^{n+1}\2^{n+1}}{(n+1)!(n+1)!\(-5)^n\2^{n+2}}\right |$

We can simplify the expression to

$L=\frac{5}{2}\lim_{n\rightarrow \infty}\left | \frac{n!(n+2)!}{(n+1)!^2}\right |=\frac{5}{2}\lim_{n\rightarrow \infty}\left | \frac{n+2}{n+1}\right |$

When we evaluate the limit, we get.

$L=\frac{5}{2}$ .

Since L>1 , we have sufficient evidence to conclude that the series is divergent.

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Example Question #11 : Ratio Test And Comparing Series

Determine if the following series is Convergent, Divergent or Neither.

$\sum_{n=0}^{\infty} \frac{(2n+3)4^{n+2}}{(n+2)5^{n+1}}$

Possible Answers:

Convergent

Not enough information.

Divergent

More tests are needed.

Neither

Correct answer:

Convergent

Explanation:

To determine if

$\sum_{n=0}^{\infty} \frac{(2n+3)4^{n+2}}{(n+2)5^{n+1}}$

is convergent, divergent or neither, we need to use the ratio test.

The ratio test is as follows.

Suppose we a series $\sum a_n$ . Then we define,

$L=\lim_{n\rightarrow \infty}\left | \frac{a_{n+1}}{a_n}\right |$ .

L<1 the series is absolutely convergent (and hence convergent).

L>1 the series is divergent.

L=1 the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply this to our situtation.

Let

$a_n=\frac{(2n+3)4^{n+2}}{(n+2)5^{n+1}}$

and

$a_{n+1}=\frac{(2(n+1)+3)4^{n+3}}{(n+3)5^{n+2}}=\frac{(2n+5)4^{n+3}}{(n+3)5^{n+2}}$

Now

$L=\lim_{n\rightarrow \infty}\left | \frac{\frac{(2n+5)4^{n+3}}{(n+3)5^{n+2}}}{\frac{(2n+3)4^{n+2}}{(n+2)5^{n+1}}}\right |$

We can rearrange the expression to be

$L=\lim_{n\rightarrow \infty}\left | \frac{(2n+5)(n+2)4^{n+3}\ 5^{n+1}}{(n+3)(2n+3)5^{n+2}\ 4^{n+2}}\right |$

We can simplify the expression to

$L=\frac{4}{5}\lim_{n\rightarrow \infty}\left | \frac{(2n+5)(n+2)}{(n+3)(2n+3)}\right |$

When we evaluate the limit, we get.

$L=\frac{4}{5}$ .

Since L<1 , we have sufficient evidence to conclude that the series is convergent.

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Example Question #2 : Comparing Series

We know that :
$chx=\frac{e^x+e^{-x}}{2}$ and $shx=\frac{e^x-e^{-x}}{2}$

We consider the series having the general term:

$v_{n}=\frac{chn}{ch2n}$

Determine the nature of the series:

$\sum_{n=1}^{\infty}v_{n}$

Possible Answers:

It will stop converging after a certain number.

$\frac{1}{2}$

The series is convergent.

The series is divergent.

$e^{-4}$

Correct answer:

The series is convergent.

Explanation:

We know that:

$chn=\frac{e^n+e^{-n}}{2}$ and therefore we deduce :

$ch2n=\frac{e^{2n }+e^{-2n}}{2}$

We will use the Comparison Test with this problem. To do this we will look at the function in general form $v_{n}\equiv \frac{e^n} {e^{2n}}$ .

We can do this since,

$e^{-n}=\frac{1}{e^n}$ and $e^{-2n}=\frac{1}{e^{2n}}$ approach zero as n approaches infinity. The limit of our function becomes,

$\lim_{n\rightarrow \infty}\frac{e^n+e^{-n}}{e^{2n}+e^{-2n}}=\frac{e^n}{e^{2n}}$

This last part gives us $v_{n}\equiv {e^{-n}}$ .

Now we know that $\frac{1}{e}< 1$ and noting that $e^{-n}}$ is a geometric series that is convergent.

We deduce by the Comparison Test that the series

having general term $v_{n}$ is convergent.

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Example Question #3 : Comparing Series

We consider the following series:

$\sum_{n=0}^{\infty} \frac{n^3+1}{n^4}$$

Determine the nature of the convergence of the series.

Possible Answers:

$\frac{1}{2}$

The series is divergent.

$\frac{3}4{}$

Correct answer:

The series is divergent.

Explanation:

We will use the comparison test to prove this result. We must note the following:

$\frac{n^3+1}{n^4}$ is positive.

We have all natural numbers n:

$n^4\ge n^3$ , this implies that

$n^4\ge n^3 \ge n$ .

Inverting we get :

$\frac{1}{n} <\frac{n^3+1}{n^4}$

Summing from 1 to $\infty$ , we have

$\sum_{n=1}^{\infty}\frac{1}{n} < \sum_{n=1}^{\infty}\frac{n^3+1}{n^4}$

We know that the $\sum_{n=1}^{\infty}\frac{1}{n}$ is divergent. Therefore by the comparison test:

$\sum_{n=1}^{\infty}\frac{n^3}{n^4+1}$

is divergent

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Example Question #9 : Comparing Series

Using the Limit Test, determine the nature of the series:

$\frac{n+1}{n^9-7}$

Possible Answers:

$\frac{1}{99}$

The series is divergent.

The series is convergent.

$\frac{1}{5}$

$\frac{2}{7}$

Correct answer:

The series is convergent.

Explanation:

We will use the Limit Comparison Test to study the nature of the series.

We note first that $n\ge 2$ , the series is positive.

We will compare the general term to $\frac{1}{n^8}$ .

We note that by letting $v_{n}=\frac{n+11}{n^9-7}$ and $u_{n}=\frac{1}{n^8}$ , we have:

$\lim_{n\rightarrow \infty}\frac{u_{n}}{v_n}}=1$ .

Therefore the two series have the same nature, (they either converge or diverge at the same time).

We will use the Integral Test to deduce that the series having the general term:

$u_{n}=\frac{1}{n^8}$ is convergent.

Note that we know that $\int_{1}^{\infty}\frac{1}{x^p}dx$ is convergent if p>1 and in our case p=8 .

This shows that the series having general term $u_{n}$ is convergent.

By the Limit Test, the series having general term v_n is convergent.

This shows that our series is convergent.

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Example Question #15 : Ratio Test And Comparing Series

We consider the following series:

$\sum_{n=0}^{\infty} \frac{n}{n^2+1}$$

Determine the nature of the convergence of the series.

Possible Answers:

$\frac{3}4{}$

The series is divergent.

$\frac{1}{2}$

Correct answer:

The series is divergent.

Explanation:

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

$\frac{n}{n^2+1}$ is positive.

We have all natural numbers n:

$n^2\ge n$ , this implies that

$n^2+1\ge n+1 > n$ .

Inverting we get :

$\frac{1}{n} <\frac{n}{n^2+1}$

Summing from 1 to $\infty$ , we have

$\sum_{n=1}^{\infty}\frac{1}{n} < \sum_{n=1}^{\infty}\frac{n}{n^2+1}$

We know that the $\sum_{n=1}^{\infty}\frac{1}{n}$ is divergent. Therefore by the Comparison Test:

$\sum_{n=1}^{\infty}\frac{n}{n^2+1}$ is divergent.

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Example Question #11 : Ratio Test And Comparing Series

Is the series

$\small \sum_{n=1}^\infty \frac{n^3}{n^7+1000}$

convergent or divergent, and why?

Possible Answers:

Divergent, by the test for divergence.

Divergent, by the comparison test.

Convergent, by the comparison test.

Divergent, by the ratio test.

Convergent, by the ratio test.

Correct answer:

Convergent, by the comparison test.

Explanation:

We will use the comparison test to prove that

$\small \sum_{n=1}^\infty \frac{n^3}{n^7+1000}$

converges (Note: we cannot use the ratio test, because then the ratio will be $\small 1$ , which means the test is inconclusive).

We will compare $\small \frac{n^3}{n^7+1000}$ to $\small \frac{1}{n^4}$ because they "behave" somewhat similarly. Both series are nonzero for all $\small n\geq 1$ , so one of the conditions is satisfied.

The series

$\small \small \sum_{n=1}^\infty \frac{1}{n^4}$

converges, so we must show that

$\small \small \frac{1}{n^4}\geq \frac{n^3}{n^4+1000}$

for $\small n\geq 1$ .

This is easy to show because

$\small \frac{1}{n^4}=\frac{n^3}{n^7}\geq \frac{n^3}{n^7+1000}$

since the denominator $\small n^7+1000$ is greater than or equal to $\small n^7$ for all $\small n\geq 1$ .

Thus, since

$\small \small \frac{1}{n^4}\geq \frac{n^3}{n^4+1000}$

and because

$\small \small \sum_{n=1}^\infty \frac{1}{n^4}$

converges, it follows that

$\small \sum_{n=1}^\infty \frac{n^3}{n^7+1000}$

converges, by comparison test.

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Example Question #1 : Ratio Test

Which of these series cannot be tested for convergence/divergence properly using the ratio test? (Which of these series fails the ratio test?)

Possible Answers:

$\sum_{k=0}^{\infty} \frac{1}{k!}$

$\sum_{k=0}^{\infty} \frac{(-2)^k}{k!}$

$\sum_{k=0}^{\infty} k$

$\sum_{k=0}^{\infty} \frac{k^2}{3^k}$

None of the other answers.

Correct answer:

$\sum_{k=0}^{\infty} k$

Explanation:

The ratio test fails when $\lim_{k\to \infty} \left |\frac{a_{k+1}}{a_k} \right | =1$ . Otherwise the series converges absolutely if $\lim_{k\to \infty} \left |\frac{a_{k+1}}{a_k} \right | <1$ , and diverges if $\lim_{k\to \infty} \left |\frac{a_{k+1}}{a_k} \right | >1$ .

Testing the series $\sum_{k=0}^{\infty} k$ , we have

$\lim_{k\to\infty} \left |\frac{a_{k+1}}{a_k} \right | = \lim_{k\to\infty} \left |\frac{(k+1)}{(k)} \right | = \lim_{k\to\infty} 1+ \frac{1}{k} = 1.$

Hence the ratio test fails here. (It is likely obvious to the reader that this series diverges already. However, we must remember that all intuition in mathematics requires rigorous justification. We are attempting that here.)

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Example Question #2 : Ratio Test

Assuming that $a_{n}=n^{\alpha}(n+1)$ , $\alpha \ge 2$ . Using the ratio test, what can we say about the series:

$\sum_{n=1}^{\infty} \frac{1}{a_{n}}$

Possible Answers:

$-\infty$

$\infty$

We cannot conclude when we use the ratio test.

It is convergent.

Correct answer:

We cannot conclude when we use the ratio test.

Explanation:

As required by this question we will have to use the ratio test. $L=\lim_{n\rightarrow \infty}\left|\frac{a_{n+1}}{a_n}\right|$ if L<1 the series converges absolutely, L>1 the series diverges, and if L=1 the series could either converge or diverge.

To do so, we will need to compute : $\frac{a_{n+1}}{a_{n}}$ . In our case:

$a_{n}=\frac{1}{a_{n}}$

Therefore

$\frac{a_{n+1}}{a_{n}}=(\frac{n}{n+1})^{\alpha}\frac{n+1}{n+2}$ .

We know that $\lim_{n\rightarrow \infty }\frac{n}{n+1}=1$

This means that

$\frac{a_{n+1}}{a_{n}}=(\frac{n}{n+1})^{\alpha}\frac{n+1}{n+2}=1\forall\quad \alpha$

Since L=1 by the ratio test, we can't conclude about the convergence of the series.

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Example Question #4 : Ratio Test

Using the ratio test,

what can we say about the series.

$\sum_{_n=1}^{\infty} \frac{1}{n^p}$ where is an integer that satisfies:

p> 1

Possible Answers:

$\frac{4}{5}$

We can't conclude when we use the ratio test.

$\infty$

We can't use the ratio test to study this series.

Correct answer:

We can't conclude when we use the ratio test.

Explanation:

Let $a_{n}$ be the general term of the series. We will use the ratio test to check the convergence of the series.

The Ratio Test states:

$L=\lim_{n\rightarrow \infty}\left|\frac{a_n+1}{a_n}\right|$

then if,

1) L<1 the series converges absolutely.

2) L>1 the series diverges.

3) L=1 the series either converges or diverges.

Therefore we need to evaluate,

$\frac{a_{n+1}}{a_{n}}$

we have,

$a_{n+1}=\frac{1}{(n+1)^p} ,a_{n}=\frac{1}{n^p}$

therefore:

$\frac{a_{n+1}}{a_{n}}=\frac{n^p}{(n+1)^p}$ .

We know that

$\lim_{n\rightarrow \infty}\frac{n}{n+1}=1$

and therefore,

$\lim_{n\rightarrow \infty}(\frac{n}{n+1})^p=1$

This means that :

$\lim_{n\rightarrow \infty}\frac{a_{n}}{a_{n+1}}=1$

By the ratio test we can't conclude about the nature of the series. We will have to use another test.

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AP Calculus BC : Series of Constants

Study concepts, example questions & explanations for AP Calculus BC

All AP Calculus BC Resources

Example Questions

Example Question #13 : Ratio Test And Comparing Series

Example Question #11 : Ratio Test And Comparing Series

Example Question #2 : Comparing Series

Example Question #3 : Comparing Series

Example Question #9 : Comparing Series

Example Question #15 : Ratio Test And Comparing Series

Example Question #11 : Ratio Test And Comparing Series

Example Question #1 : Ratio Test

Example Question #2 : Ratio Test

Example Question #4 : Ratio Test

All AP Calculus BC Resources

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