Stewart Section 11.4: The Comparison Tests

46 questions

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Stewart Section 11.4: The Comparison Tests 0/46
1 Comparison Test - Concepts · Level 2
Suppose \(\sum a_n\) and \(\sum b_n\) are series with positive terms and \(\sum b_n\) is known to be convergent.
(a) If \(a_n > b_n\) for all \(n\), what can you say about \(\sum a_n\)? Why?
(b) If \(a_n < b_n\) for all \(n\), what can you say about \(\sum a_n\)? Why?

Enter your answer directly below each part above.

2 Comparison Test - Concepts · Level 2
Suppose \(\sum a_n\) and \(\sum b_n\) are series with positive terms and \(\sum b_n\) is known to be divergent.
(a) If \(a_n > b_n\) for all \(n\), what can you say about \(\sum a_n\)? Why?
(b) If \(a_n < b_n\) for all \(n\), what can you say about \(\sum a_n\)? Why?

Enter your answer directly below each part above.

3 Comparison Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^3 + 8} \)
4 Comparison Test · Level 2
\( \displaystyle\sum_{n=2}^{\infty} \dfrac{1}{\sqrt{n} - 1} \)
5 Comparison Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{n + 1}{n \sqrt{n}} \)
6 Comparison Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{n - 1}{n^3 + 1} \)
7 Comparison Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{9^n}{3 + 10^n} \)
8 Comparison Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{6^n}{5^n - 1} \)
9 Comparison Test · Level 2
\( \displaystyle\sum_{k=1}^{\infty} \dfrac{\ln k}{k} \)
10 Comparison Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{k \sin^2 k}{1 + k^3} \)
11 Comparison Test · Level 3
\( \displaystyle\sum_{k=1}^{\infty} \dfrac{\sqrt[3]{k}}{\sqrt{k^3 + 4k + 3}} \)
12 Comparison Test · Level 3
\( \displaystyle\sum_{k=1}^{\infty} \dfrac{(2k - 1)(k^2 - 1)}{(k + 1)(k^2 + 4)^2} \)
13 Comparison Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{1 + \cos n}{e^n} \)
14 Comparison Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{\sqrt[3]{3n^4 + 1}} \)
15 Comparison Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{4^{n+1}}{3^n - 2} \)
16 Comparison Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^n} \)
17 Comparison Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{\sqrt{n^2 + 1}} \)
18 Comparison Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{2}{\sqrt{n} + 2} \)
19 Comparison Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{n + 1}{n^3 + n} \)
20 Comparison Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{n^2 + n + 1}{n^4 + n^2} \)
21 Comparison Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{\sqrt{1 + n}}{2 + n} \)
22 Comparison Test · Level 2
\( \displaystyle\sum_{n=3}^{\infty} \dfrac{n + 2}{(n + 1)^3} \)
23 Comparison Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{5 + 2n}{(1 + n^2)^2} \)
24 Comparison Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{n + 3^n}{n + 2^n} \)
25 Comparison Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{e^n + 1}{n e^n + 1} \)
26 Comparison Test · Level 3
\( \displaystyle\sum_{n=2}^{\infty} \dfrac{1}{n \sqrt{n^2 - 1}} \)
27 Comparison Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \left(1 + \dfrac{1}{n}\right)^2 e^{-n} \)
28 Comparison Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{e^{\dfrac{1}{n}}}{n} \)
29 Comparison Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n!} \)
30 Comparison Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{n^n} \)
31 Comparison Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \sin\left(\dfrac{1}{n}\right) \)
32 Comparison Test · Level 4
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{1 + \dfrac{1}{n}}} \)
33 Estimating Sums · Level 3
Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{5 + n^5}\)
34 Estimating Sums · Level 3
Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{e^{\dfrac{1}{n}}}{n^4}\)
35 Estimating Sums · Level 3
Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. \(\displaystyle\sum_{n=1}^{\infty} 5^{-n} \cos^2 n\)
36 Estimating Sums · Level 3
Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{3^n + 4^n}\)
37 Comparison Test - Proof · Level 3
The meaning of the decimal representation of a number \(0.d_1 d_2 d_3 \ldots\) (where the digit \(d_i\) is one of the numbers 0, 1, 2, ...,
9) is that \(0.d_1 d_2 d_3 d_4 \ldots = \dfrac{d_1}{10} + \dfrac{d_2}{10^2} + \dfrac{d_3}{10^3} + \dfrac{d_4}{10^4} + \cdots\) Show that this series always converges.
38 Comparison Test - p-values · Level 3
For what values of \(p\) does the series \(\displaystyle\sum_{n=2}^{\infty} 1/(n^p \ln n)\) converge?
39 Comparison Test - Proof · Level 4
Prove that if \(a_n \geq 0\) and \(\sum a_n\) converges, then \(\sum a_n^2\) also converges.
40 Limit Comparison Test - Proof · Level 4
(a) Suppose that \(\sum a_n\) and \(\sum b_n\) are series with positive terms and \(\sum b_n\) is convergent. Prove that if \(\operatorname*{lim}\limits_{n \rightarrow \infty} \dfrac{a_n}{b_n} = 0\) then \(\sum a_n\) is also convergent.
(b) Use part (a) to show that the series converges. (i) \(\displaystyle\sum_{n=1}^{\infty} \dfrac{\ln n}{n^3}\) (ii) \(\displaystyle\sum_{n=1}^{\infty} \dfrac{\ln n}{\sqrt{n} e^n}\)

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41 Limit Comparison Test - Proof · Level 4
(a) Suppose that \(\sum a_n\) and \(\sum b_n\) are series with positive terms and \(\sum b_n\) is divergent. Prove that if \(\operatorname*{lim}\limits_{n \rightarrow \infty} \dfrac{a_n}{b_n} = \infty\) then \(\sum a_n\) is also divergent.
(b) Use part (a) to show that the series diverges. (i) \(\displaystyle\sum_{n=2}^{\infty} \dfrac{1}{\ln n}\) (ii) \(\displaystyle\sum_{n=1}^{\infty} \dfrac{\ln n}{n}\)

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42 Comparison Test - Proof · Level 4
Give an example of a pair of series \(\sum a_n\) and \(\sum b_n\) with positive terms where \(\operatorname*{lim}\limits_{n \rightarrow \infty} \left(\dfrac{a_n}{b_n}\right) = 0\) and \(\sum b_n\) diverges, but \(\sum a_n\) converges. (Compare with Exercise 40.)
43 Comparison Test - Proof · Level 4
Show that if \(a_n > 0\) and \(\operatorname*{lim}\limits_{n \rightarrow \infty} n a_n \neq 0\), then \(\sum a_n\) is divergent.
44 Comparison Test - Proof · Level 4
Show that if \(a_n > 0\) and \(\sum a_n\) is convergent, then \(\sum \ln(1 + a_n)\) is convergent.
45 Comparison Test - Proof · Level 4
If \(\sum a_n\) is a convergent series with positive terms, is it true that \(\sum \sin(a_n)\) is also convergent?
46 Comparison Test - Proof · Level 4
If \(\sum a_n\) and \(\sum b_n\) are both convergent series with positive terms, is it true that \(\sum a_n b_n\) is also convergent?

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