Stewart Section 11.3: The Integral Test

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Stewart Section 11.3: The Integral Test 0/42
1 Integral Test - Concepts · Level 2
Draw a picture to show that \(\displaystyle\sum_{n=2}^{\infty} \dfrac{1}{n^{1.3}} < \displaystyle\int_{1}^{\infty} \dfrac{1}{x^{1.3}} d x\) What can you conclude about the series?
2 Integral Test - Concepts · Level 2
Suppose \(f\) is a continuous positive decreasing function for \(x \geq 1\) and \(a_n = f(n)\). By drawing a picture, rank the following three quantities in increasing order: \(\displaystyle\int_{1}^{6} f(x) d x\), \(\displaystyle\sum_{i=1}^5 a_i\), \(\displaystyle\sum_{i=2}^6 a_i\)
3 Integral Test · Level 2
Use the Integral Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} n^{-3}\)
4 Integral Test · Level 2
Use the Integral Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} n^{-0.3}\)
5 Integral Test · Level 2
Use the Integral Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{2}{5n - 1}\)
6 Integral Test · Level 2
Use the Integral Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{(3n - 1)^4}\)
7 Integral Test · Level 2
Use the Integral Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n}{n^2 + 1}\)
8 Integral Test · Level 3
Use the Integral Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} n^2 e^{-n^3}\)
9 Series - Convergence · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{\sqrt{2}}} \)
10 Series - Convergence · Level 2
\( \displaystyle\sum_{n=2}^{\infty} n^{-0.9999} \)
11 Series - Convergence · Level 2
\( 1 + \dfrac{1}{8} + \dfrac{1}{27} + \dfrac{1}{64} + \dfrac{1}{125} + \cdots \)
12 Series - Convergence · Level 2
\( \dfrac{1}{5} + \dfrac{1}{7} + \dfrac{1}{9} + \dfrac{1}{11} + \dfrac{1}{13} + \cdots \)
13 Series - Convergence · Level 2
\( \dfrac{1}{3} + \dfrac{1}{7} + \dfrac{1}{11} + \dfrac{1}{15} + \dfrac{1}{19} + \cdots \)
14 Series - Convergence · Level 2
\( 1 + \dfrac{1}{2 \sqrt{2}} + \dfrac{1}{3 \sqrt{3}} + \dfrac{1}{4 \sqrt{4}} + \dfrac{1}{5 \sqrt{5}} + \cdots \)
15 Series - Convergence · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{\sqrt{n} + 4}{n^2} \)
16 Series - Convergence · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{\sqrt{n}}{1 + n^{\dfrac{3}{2}}} \)
17 Series - Convergence · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 4} \)
18 Series - Convergence · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 2n + 2} \)
19 Series - Convergence · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{n^3}{n^4 + 4} \)
20 Series - Convergence · Level 3
\( \displaystyle\sum_{n=3}^{\infty} \dfrac{3n - 4}{n^2 - 2n} \)
21 Series - Convergence · Level 3
\( \displaystyle\sum_{n=2}^{\infty} \dfrac{1}{n \ln n} \)
22 Series - Convergence · Level 3
\( \displaystyle\sum_{n=2}^{\infty} \dfrac{\ln n}{n^2} \)
23 Series - Convergence · Level 3
\( \displaystyle\sum_{k=1}^{\infty} k e^{-k} \)
24 Series - Convergence · Level 3
\( \displaystyle\sum_{k=1}^{\infty} k e^{-k^2} \)
25 Series - Convergence · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + n^3} \)
26 Series - Convergence · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{n}{n^4 + 1} \)
27 Integral Test - Concepts · Level 2
Explain why the Integral Test can't be used to determine whether the series is convergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{\cos \pi n}{\sqrt{n}}\)
28 Integral Test - Concepts · Level 2
Explain why the Integral Test can't be used to determine whether the series is convergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{\cos^2 n}{1 + n^2}\)
29 Series - p-values · Level 3
Find the values of \(p\) for which the series is convergent. \(\displaystyle\sum_{n=2}^{\infty} \dfrac{1}{n (\ln n)^p}\)
30 Series - p-values · Level 4
Find the values of \(p\) for which the series is convergent. \(\displaystyle\sum_{n=3}^{\infty} \dfrac{1}{n \ln n [\ln(\ln n)]^p}\)
31 Series - p-values · Level 3
Find the values of \(p\) for which the series is convergent. \(\displaystyle\sum_{n=1}^{\infty} n(1 + n^2)^p\)
32 Series - p-values · Level 3
Find the values of \(p\) for which the series is convergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{\ln n}{n^p}\)
33 Riemann Zeta Function · Level 3
The Riemann zeta-function \(\zeta\) is defined by \(\zeta(x) = \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^x}\) and is used in number theory to study the distribution of prime numbers. What is the domain of \(\zeta\)?
34 Riemann Zeta Function · Level 3
Leonhard Euler was able to calculate the exact sum of the \(p\)-series with \(p = 2\): \(\zeta(2) = \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2} = \dfrac{\pi^2}{6}\) Use this fact to find the sum of each series.
(a) \(\displaystyle\sum_{n=2}^{\infty} \dfrac{1}{n^2}\)
(b) \(\displaystyle\sum_{n=3}^{\infty} \dfrac{1}{(n+1)^2}\)
(c) \(\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{(2n)^2}\)

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35 Riemann Zeta Function · Level 3
Euler also found the sum of the \(p\)-series with \(p = 4\): \(\zeta(4) = \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^4} = \dfrac{\pi^4}{90}\) Use Euler's result to find the sum of the series.
(a) \(\displaystyle\sum_{n=1}^{\infty} \left(\dfrac{3}{n}\right)^4\)
(b) \(\displaystyle\sum_{k=5}^{\infty} \dfrac{1}{(k - 2)^4}\)

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36 Remainder Estimates · Level 3
(a) Find the partial sum \(s_{10}\) of the series \(\displaystyle\sum_{n=1}^{\infty} 1/n^4\). Estimate the error in using \(s_{10}\) as an approximation to the sum of the series.
(b) Use the remainder estimate with \(n = 10\) to give an improved estimate of the sum.
(c) Compare your estimate in part (b) with the exact value given in Exercise 35.
(d) Find a value of \(n\) so that \(s_n\) is within 0.00001 of the sum.

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37 Remainder Estimates · Level 3
(a) Use the sum of the first 10 terms to estimate the sum of the series \(\displaystyle\sum_{n=1}^{\infty} 1/n^2\). How good is this estimate?
(b) Improve this estimate using the remainder estimate with \(n = 10\).
(c) Compare your estimate in part (b) with the exact value given in Exercise 34.
(d) Find a value of \(n\) that will ensure that the error in the approximation \(s \approx s_n\) is less than 0.001.

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38 Remainder Estimates · Level 3
Find the sum of the series \(\displaystyle\sum_{n=1}^{\infty} n e^{-2n}\) correct to four decimal places.
39 Remainder Estimates · Level 3
Estimate \(\displaystyle\sum_{n=1}^{\infty} (2n + 1)^{-6}\) correct to five decimal places.
40 Remainder Estimates · Level 3
How many terms of the series \(\displaystyle\sum_{n=2}^{\infty} 1/[n (\ln n)^2]\) would you need to add to find its sum to within 0.01?
41 Remainder Estimates · Level 4
Show that if we want to approximate the sum of the series \(\displaystyle\sum_{n=1}^{\infty} n^{-1.001}\) so that the error is less than 5 in the ninth decimal place, then we need to add more than \(10^{11.301}\) terms!
42 Remainder Estimates · Level 3
(a) Show that the series \(\displaystyle\sum_{n=1}^{\infty} (\ln n)^2 / n^2\) is convergent.
(b) Find an upper bound for the error in the approximation \(s \approx s_n\).
(c) What is the smallest value of \(n\) such that this upper bound is less than 0.05?
(d) Find \(s_n\) for this value of \(n\).

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