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Calculus 2 : Comparing Series

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Example Questions

Example Question #11 : 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:

$\frac{1}{2}$

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:

$\frac{n}{n^2+1}$ 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 #162 : Series In Calculus

Is the series

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

convergent or divergent, and why?

Possible Answers:

Divergent, by the comparison test.

Convergent, by the ratio test.

Convergent, by the comparison test.

Divergent, by the test for divergence.

Divergent, 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 #2 : P Series

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

$\sum_{n = 1}^{\infty } \frac{1}{n^{3} + 2}$

Possible Answers:

Converges

Conditionally converges.

There is not enough information to decide convergence.

Diverges

Neither converges nor diverges.

Correct answer:

Converges

Explanation:

We can compare this to the series $\sum_{n = 1}^{\infty } \frac{1}{n^{3}}$ which we know converges by the p-series test.

To figure this out, let's first compare $n^{3} + 2$ to $n^{3}$ . For any number n, $n^{3} + 2$ will be larger than $n^{3}$ .

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, $n^{3} + 2 > n^{3}}$ turns into

$\frac{1}{n^{3}} > \frac{1}{n^{3} + 2}$ .

And so, because $\frac{1}{n ^{3}}$ converges, thus our series also converges.

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Example Question #1 : Comparing Rates Of Convergence

For which values of p is

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

convergent?

Possible Answers:

All positive values of

p=2

it doesn't converge for any values of

only p > 1

Correct answer:

only p > 1

Explanation:

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

$\int_1^{\infty} \frac{1}{x^p} dx$

converges, then our series converges.

We can rewrite the integral as

$\int x^{-p} dx$

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

$\frac{x^{1-p}}{1-p} \Big|_{x=1}^\infty$ .

We know that this only goes to zero if 1-p < 0 . Subtracting p from both sides, we get

1 < p .

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Example Question #11 : Comparing Series

$\sum_{n=3}^{\infty}\frac{1}{n^2+2}$

Determine the convergence of the series using the Comparison Test.

Possible Answers:

Series diverges

Cannot be determined

Series converges

Correct answer:

Series converges

Explanation:

We compare this series to the series $\sum_{n=3}^{\infty}\frac{1}{n^2}$

Because

n^2+2>n^2 for n=3,4,5,...

it follows that

$\frac{1}{n^2+2}<\frac{1}{n^2}$ for n=3,4,5,...

This implies

$\sum_{n=3}^{\infty}\frac{1}{n^2+2}<\sum_{n=3}^{\infty}\frac{1}{n^2}$

Because the series on the right has an exponent 2>1 ,

the series on the right converges

$\sum_{n=3}^{\infty}\frac{1}{n^2+2}<\sum_{n=3}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}-\frac{5}{4}$

making

$\sum_{n=3}^{\infty}\frac{1}{n^2+2}$

converge as well.

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Example Question #2951 : Calculus Ii

$\sum_{n=2}^{\infty}\frac{1}{ln\,n}$

Determine the convergence of the series using the Comparison Test.

Possible Answers:

Series converges

Series diverges

Cannot be determined

Correct answer:

Series diverges

Explanation:

We compare this series to the series $\sum_{n=2}^{\infty}\frac{1}{n}$

Because

$ln\,n<n$ for n=2,3,4,...

it follows that

$\frac{1}{ln\,n}>\frac{1}{n}$ for n=2,3,4,...

This implies

$\sum_{n=2}^{\infty}\frac{1}{ln\,n}>\sum_{n=2}^{\infty}\frac{1}{n}$

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

the series on the right diverges

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

making

$\sum_{n=2}^{\infty}\frac{1}{ln\,n}$

diverge as well.

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Example Question #2952 : Calculus Ii

Does the series $\sum_{n=1}^\infty\frac{1}{n(n+1)}$ converge or diverge? If it does converge, then what value does it converge to?

Possible Answers:

Diverges

Converges to 1

Converges to $\sqrt2$

Converges to $\frac{1}{2}$

Converges to $\frac{\sqrt2}{2}$

Correct answer:

Converges to 1

Explanation:

To show this series converges, we use direct comparison with

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

which converges by the p-series test with p=2>1 .

Thus we must show that

$\frac{1}{n(n+1)}\le\frac{1}{n^2}$ .

Cross multiplying the previous section and multiplying by n+1 , we obtain $n^2\le{n^2+n}\leftrightarrow{n\ge0}$ .

Since this holds for all n=1,2,3,... we can conclude that

$\frac{1}{n(n+1)}\le\frac{1}{n^2}$ .

Summing from to $\infty$ , and noting that

$\frac{1}{n(n+1)}>0$ for all n=1,2,3,... , we obtain the following inequality:

$0<\sum_{n=1}^\infty\frac{1}{n(n+1)}<\sum_{n=1}^\infty\frac{1}{n^2}$ .

Therefore the series

$\sum_{n=1}^\infty\frac{1}{n(n+1)}$

converges by direct comparison.

Now to find the value, we note that

$\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$ ,

so that

$\sum_{n=1}^\infty\frac{1}{n(n+1)}=\sum_{n=1}^\infty\biggr(\frac{1}{n}-\frac{1}{n+1}\biggr)$ .

Now let

$s_k=\sum_{n=1}^k\biggr(\frac{1}{n}-\frac{1}{n+1}\biggr)$

be a sequence of partial sums.

Then we have $s_k=\sum_{n=1}^k\biggr(\frac{1}{n}-\frac{1}{n+1}\biggr)$

$=\biggr(1-\frac{1}{2}\biggr)+\biggr(\frac{1}{2}-\frac{1}{3}\biggr)+\biggr(\frac{1}{3}-\frac{1}{4}\biggr)+...+\biggr(\frac{1}{k-1}-\frac{1}{k}\biggr)$

$=1-\frac{1}{k}$

Therefore

$s_k=1-\frac{1}{k}$ .

Taking the limit as $k\rightarrow\infty$ , we obtain the following:

$\lim_{k\rightarrow\infty}s_k=\lim_{k\rightarrow\infty}\sum_{n=0}^k\frac{1}{n(n+1)}=\sum_{n=0}^\infty\frac{1}{n(n+1)}=\lim_{k\rightarrow\infty}\biggr(1-\frac{1}{k}\biggr)=1-\frac{1}{\infty}=1$

Therefore we have

$\sum_{n=0}^\infty\frac{1}{n(n+1)}=1$ .

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Example Question #2951 : Calculus Ii

Does the series converge?

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

Possible Answers:

Yes

Cannot be determined

Correct answer:

Explanation:

Notice that n-1<n for n=2,3,...

This implies that

$\frac{1}{n-1}>\frac{1}{n}$ for n=2,3,...

Which then implies

$\sum_{n=2}^{\infty}\frac{1}{n-1}>\sum_{n=2}^{\infty}\frac{1}{n}$

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

$\sum_{n=2}^{\infty}\frac{1}{n-1}> \infty$

and thus the series does NOT converge.

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Example Question #2952 : Calculus Ii

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

$\sum_{n=0}^{\infty}\sqrt{n+1}-\sqrt{n}$

Possible Answers:

Converges in an interval

Converges absolutely

Does not converge at all

Converges conditionally

Correct answer:

Converges absolutely

Explanation:

Untitled

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Example Question #2953 : Calculus Ii

Determine whether the series converges

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

Possible Answers:

Converges absolutely

Converges conditionally

Does not converge at all

Converges in an interval

Correct answer:

Converges absolutely

Explanation:

Untitled

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Calculus 2 : Comparing Series

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #11 : Comparing Series

Example Question #162 : Series In Calculus

Example Question #2 : P Series

Example Question #1 : Comparing Rates Of Convergence

Example Question #11 : Comparing Series

Example Question #2951 : Calculus Ii

Example Question #2952 : Calculus Ii

Example Question #2951 : Calculus Ii

Example Question #2952 : Calculus Ii

Example Question #2953 : Calculus Ii

All Calculus 2 Resources

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