Calculus: Early Transcendental Functions 6th Edition solutions

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{1}{n+3}\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{2}{3 n+5}\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{1}{2^{n}}\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} 3^{-n}\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} e^{-n}\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} n e^{-n / 2}\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\frac{1}{2}+\frac{1}{5}+\frac{1}{10}+\frac{1}{17}+\frac{1}{26}+\cdots\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{9}+\frac{1}{11}+\cdots\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\frac{\ln 2}{2}+\frac{\ln 3}{3}+\frac{\ln 4}{4}+\frac{\ln 5}{5}+\frac{\ln 6}{6}+\cdots\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\frac{\ln 2}{\sqrt{2}}+\frac{\ln 3}{\sqrt{3}}+\frac{\ln 4}{\sqrt{4}}+\frac{\ln 5}{\sqrt{5}}+\frac{\ln 6}{\

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\begin{array}{l} \frac{1}{\sqrt{1}(\sqrt{1}+1)}+\frac{1}{\sqrt{2}(\sqrt{2}+1)}+\frac{1}{\sqrt{3}(\sqrt{3}

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\frac{1}{4}+\frac{2}{7}+\frac{3}{12}+\cdots+\frac{n}{n^{2}+3}+\cdots\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{\arctan n}{n^{2}+1}\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\sum_{n=2}^{\infty} \frac{\ln n}{n^{3}}\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{\ln n}{n^{2}}\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\sum_{n=2}^{\infty} \frac{1}{n \sqrt{\ln n}}\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{1}{(2 n+3)^{3}}\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{n+2}{n+1}\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{4 n}{2 n^{2}+1}\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{1}{\sqrt{n+2}}\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{n}{n^{4}+1}\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 1-22, confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{n}{n^{4}+2 n^{2}+1}\)

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 23 and 24, use the Integral Test to determine the convergence or divergence of the series, where k is a positive integer. \(\sum_{n=1}^{\infty} \frac{n^{k-1}}{n^{k}+c}\) Text Transcri

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 23 and 24, use the Integral Test to determine the convergence or divergence of the series, where k is a positive integer. \(\sum_{n=1}^{\infty} n^{k} e^{-n}\) Text Transcription: su

Calculus: Early Transcendental Functions 6
Requirements of the Integral Test In Exercises 25-28, explain why the Integral Test does not apply to the series. \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}\) Text Transcription: sum_n=1^infinity (-1)^n/n

Calculus: Early Transcendental Functions 6
Requirements of the Integral Test In Exercises 25-28, explain why the Integral Test does not apply to the series. \(\sum_{n=1}^{\infty} e^{-n} \cos n\) Text Transcription: sum_n=1^infinity e^-n cos n

Calculus: Early Transcendental Functions 6
Requirements of the Integral Test In Exercises 25-28, explain why the Integral Test does not apply to the series. \(\sum_{n=1}^{\infty} \frac{2+\sin n}{n}\) Text Transcription: sum_n=1^infinity 2+sin n/n

Calculus: Early Transcendental Functions 6
Requirements of the Integral Test In Exercises 25-28, explain why the Integral Test does not apply to the series. \(\sum_{n=1}^{\infty}\left(\frac{\sin n}{n}\right)^{2}\) Text Transcription: sum_n=1^infinity (sin n/n)^

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 29-32, use the Integral Test to determine the convergence or divergence of the p-series. \(\sum_{n=1}^{\infty} \frac{1}{n^{3}}\) Text Transcription: sum_n=1^infinity 1/n^3

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 29-32, use the Integral Test to determine the convergence or divergence of the p-series. \(\sum_{n=1}^{\infty} \frac{1}{n^{1 / 2}}\) Text Transcription: sum_n=1^infinity 1/n^1/2

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 29-32, use the Integral Test to determine the convergence or divergence of the p-series. \(\sum_{n=1}^{\infty} \frac{1}{n^{1 / 4}}\) Text Transcription: sum_n=1^infinity 1/n^1/4

Calculus: Early Transcendental Functions 6
Using the Integral Test In Exercises 29-32, use the Integral Test to determine the convergence or divergence of the p-series. \(\sum_{n=1}^{\infty} \frac{1}{n^{5}}\) Text Transcription: sum_n=1^infinity 1/n^5

Calculus: Early Transcendental Functions 6
Using a p-Series In Exercises 33-38, use Theorem 9.11 to determine the convergence or divergence of the p-series. \(\sum_{n=1}^{\infty} \frac{1}{\sqrt[5]{n}}\) Text Transcription: sum_n=1^infinity 1/5 sqrt n

Calculus: Early Transcendental Functions 6
Using a p-Series In Exercises 33-38, use Theorem 9.11 to determine the convergence or divergence of the p-series. \(\sum_{n=1}^{\infty} \frac{3}{n^{5 / 3}}\) Text Transcription: sum_n=1^infinity 3/n^5/3

Calculus: Early Transcendental Functions 6
Using a p-Series In Exercises 33-38, use Theorem 9.11 to determine the convergence or divergence of the p-series. \(1+\frac{1}{2 \sqrt{2}}+\frac{1}{3 \sqrt{3}}+\frac{1}{4 \sqrt{4}}+\frac{1}{5 \sqrt{5}}+\cdots\) Text Transc

Calculus: Early Transcendental Functions 6
Using a p-Series In Exercises 33-38, use Theorem 9.11 to determine the convergence or divergence of the p-series. \(1+\frac{1}{\sqrt[3]{4}}+\frac{1}{\sqrt[3]{9}}+\frac{1}{\sqrt[3]{16}}+\frac{1}{\sqrt[3]{25}}+\cdots\) Text

Calculus: Early Transcendental Functions 6
Using a p-Series In Exercises 33-38, use Theorem 9.11 to determine the convergence or divergence of the p-series. \(\sum_{n=1}^{\infty} \frac{1}{n^{1.04}}\) Text Transcription: sum_n=1^infinity 1/n^1.04

Calculus: Early Transcendental Functions 6
Using a p-Series In Exercises 33-38, use Theorem 9.11 to determine the convergence or divergence of the p-series. \(\sum_{n=1}^{\infty} \frac{1}{n^{n}}\) Text Transcription: sum_n=1^infinity 1/n^pi

Calculus: Early Transcendental Functions 6
Numerical and Graphical Analysis Use a graphing utility to find the indicated partial sum \(S_n\) and complete the table. Then use a graphing utility to graph the first 10 terms of the sequence of partial sums. For each series, compare the rate at which the sequence of partial sums approaches the sum

Calculus: Early Transcendental Functions 6
Numerical Reasoning Because the harmonic series diverges, it follows that for any positive real number M there exists a positive integer N such that the partial sum \(\sum_{n=1}^{N} \frac{1}{n}>M\) (a) Use a graphing utility to complete the table. (b) As the real number M increas

Calculus: Early Transcendental Functions 6
Integral Test State the Integral Test and give an example of its use.

Calculus: Early Transcendental Functions 6
p-Series Define a series and state the requirements for its convergence.

Calculus: Early Transcendental Functions 6
Using a Series A friend in your calculus class tells you that the following series converges because the terms are very small and approach 0 rapidly. Is your friend correct? Explain. \(\frac{1}{10,000}+\frac{1}{10,001}+\frac{1}{10,002}+\cdots\)

Calculus: Early Transcendental Functions 6
Using a Function Let f be a positive,continuous, and decreasing function for \(x \geq 1,\) such that \(a_{n}=f(n)\). Use a graph to rank the following quantities in decreasing order. Explain your reasoning. (a) \(\sum_{n=2}^{7} a_{n}\) (b) \(\int_{1}^{7} f(x) d x\) (c) \(\sum_{n=1}^{6} a_{

Calculus: Early Transcendental Functions 6
Using a Series Use a graph to show that the inequality is true. What can you conclude about the convergence or divergence of the series? Explain. (a) \(\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}>\int_{1}^{\infty} \frac{1}{\sqrt{x}} d x\) (b) \(\sum_{n=2}^{\infty} \frac{1}{n^{2}}<\int_{1}^{\infty

Calculus: Early Transcendental Functions 6
HOW DO YOU SEE IT? The graphs show the sequences of partial sums of the p-series \(\sum_{n=1}^{\infty} \frac{1}{n^{0.4}} \quad \text { and } \sum_{n=1}^{\infty} \frac{1}{n^{1.5}}\) Using Theorem 9.11, the first series diverges and the second series converges. Explain how the graphs show this

Calculus: Early Transcendental Functions 6
Finding Values In Exercises 47-52, find the positive values of p for which the series converges. \(\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{p}}\) Text Transcription: sum_n=2^infinity 1/n(ln n)^p

Calculus: Early Transcendental Functions 6
Finding Values In Exercises 47-52, find the positive values p of for which the series converges. \(\sum_{n=2}^{\infty} \frac{\ln n}{n^{p}}\) Text Transcription: sum_n=2^infinity ln n/n^p

Calculus: Early Transcendental Functions 6
Finding Values In Exercises 47-52, find the positive values p of for which the series converges. \(\sum_{n=1}^{\infty} \frac{n}{\left(1+n^{2}\right)^{p}}\) Text Transcription: sum_n=1^infinity n/(1+n^2)^p

Calculus: Early Transcendental Functions 6
Finding Values In Exercises 47-52, find the positive values p of for which the series converges. \(\sum_{n=1}^{\infty} n\left(1+n^{2}\right)^{p}\) Text Transcription: sum_n=1^infinity n(1+n^2)^p

Calculus: Early Transcendental Functions 6
Finding Values In Exercises 47-52, find the positive values p of for which the series converges. \(\sum_{n=1}^{\infty}\left(\frac{3}{p}\right)^{n}\) Text Transcription: sum_n=1^infinity (3/p)^n

Calculus: Early Transcendental Functions 6
Finding Values In Exercises 47-52, find the positive values p of for which the series converges. \(\sum_{n=3}^{\infty} \frac{1}{n \ln n[\ln (\ln n)]^{p}}\) Text Transcription: sum_n=3^infinity 1/n ln n[ln(ln n)]^p

Calculus: Early Transcendental Functions 6
Proof Let f be a positive, continuous, and decreasing function for \(x \geq 1\), such that \(a_{n}=f(n)\). Prove that if the series \(\sum_{n=1}^{\infty} a_{n}\) Converges to S, then the remainder \(R_{N}=S-S_{N}\) is bounded by \(0 \leq R_{N} \leq \int_{N}^{\infty} f(x) d x\)

Calculus: Early Transcendental Functions 6
Using a Remainder Show that the result of Exercise 53 can be written as \(\sum_{n=1}^{N} a_{n} \leq \sum_{n=1}^{\infty} a_{n} \leq \sum_{n=1}^{N} a_{n}+\int_{N}^{\infty} f(x) d x\) Text Transcription: sum_n=1^N a_n leq

Calculus: Early Transcendental Functions 6
Approximating a Sum In Exercises 55-60, use the result of Exercise 53 to approximate the sum of the convergent series using the indicated number of terms. Include an estimate of the maximum error for your approximation. \(\sum_{n=1}^{\infty} \frac{1}{n^{2}} \text {, five terms }\)

Calculus: Early Transcendental Functions 6
Approximating a Sum In Exercises 55-60, use the result of Exercise 53 to approximate the sum of the convergent series using the indicated number of terms. Include an estimate of the maximum error for your approximation. \(\sum_{n=1}^{\infty} \frac{1}{n^{5}}, \operatorname{six} \text { terms }\)

Calculus: Early Transcendental Functions 6
Approximating a Sum In Exercises 55-60, use the result of Exercise 53 to approximate the sum of the convergent series using the indicated number of terms. Include an estimate of the maximum error for your approximation. \(\sum_{n=1}^{\infty} \frac{1}{n^{2}+1}, \text { ten terms }\)

Calculus: Early Transcendental Functions 6
Approximating a Sum In Exercises 55-60, use the result of Exercise 53 to approximate the sum of the convergent series using the indicated number of terms. Include an estimate of the maximum error for your approximation. \(\sum_{n=1}^{\infty} \frac{1}{(n+1)[\ln (n+1)]^{3}} \text {, ten terms }\)

Calculus: Early Transcendental Functions 6
Approximating a Sum In Exercises 55-60, use the result of Exercise 53 to approximate the sum of the convergent series using the indicated number of terms. Include an estimate of the maximum error for your approximation. \(\sum_{n=1}^{\infty} n e^{-n^{2}} \text {, four terms }\)

Calculus: Early Transcendental Functions 6
Approximating a Sum In Exercises 55-60, use the result of Exercise 53 to approximate the sum of the convergent series using the indicated number of terms. Include an estimate of the maximum error for your approximation. \(\sum_{n=1}^{\infty} e^{-n} \text {, four terms }\)

Calculus: Early Transcendental Functions 6
Finding a Value In Exercises 61-64, use the result of Exercise 53 to find N such that \(R_{N} \leq 0.001\) for the convergent series. \(\sum_{n=1}^{\infty} \frac{1}{n^{4}}\) Text Transcription: R_N leq 0.001 sum_n=1^in

Calculus: Early Transcendental Functions 6
Finding a Value In Exercises 61-64, use the result of Exercise 53 to find N such that \(R_{N} \leq 0.001\) for the convergent series. \(\sum_{n=1}^{\infty} \frac{1}{n^{3 / 2}}\) Text Transcription: R_N leq 0.001 sum_n=

Calculus: Early Transcendental Functions 6
Finding a Value In Exercises 61-64, use the result of Exercise 53 to find N such that \(R_{N} \leq 0.001\) for the convergent series. \(\sum_{n=1}^{\infty} e^{-n / 2}\) Text Transcription: R_N leq 0.001 sum_n=1^infin

Calculus: Early Transcendental Functions 6
Finding a Value In Exercises 61-64, use the result of Exercise 53 to find N such that \(R_{N} \leq 0.001\) for the convergent series. \(\sum_{n=1}^{\infty} \frac{1}{n^{2}+1}\) Text Transcription: R_N leq 0.001 sum_n=1^

Calculus: Early Transcendental Functions 6
Comparing Series (a) Show that \(\sum_{n=2}^{\infty} \frac{1}{n^{1.1}}\) converges and \(\sum_{n=2}^{\infty} \frac{1}{n \ln n}\) diverges. (b) Compare the first five terms of each series in part (a). (c) Find n > 3 such that \(\frac{1}{n^{1.1}}<\frac{1}{n \ln n}\).

Calculus: Early Transcendental Functions 6
Using a p-Series Ten terms are used to approximate a convergent series. Therefore, the remainder is a function of and is a function of p and is \(0 \leq R_{10}(p) \leq \int_{10}^{\infty} \frac{1}{x^{p}} d x, \quad p>1\) (a) Perform the integration in the inequality. (b) Use a graphing ut

Calculus: Early Transcendental Functions 6
Eulers Contact Let \(S_{n}=\sum_{k=1}^{n} \frac{1}{k}=1+\frac{1}{2}+\cdots+\frac{1}{n}\) (a) Show that \(\ln (n+1) \leq S_{n} \leq 1+\ln n\) (b) Show that the sequence \(\left\{a_{n}\right\}=\left\{S_{n}-\ln n\right\}\) (c) Show that the sequence \(\left\{a_{n}\right\}\) is decreasi

Calculus: Early Transcendental Functions 6
Finding a Sum Find the sum of the series \(\sum_{n=2}^{\infty} \ln \left(1-\frac{1}{n^{2}}\right)\) Text Transcription: sum_n=2^infinity ln(1-1/n^2)

Calculus: Early Transcendental Functions 6
Using a Series Consider the series \(\sum_{n=2}^{\infty} x^{\ln n}\) (a) Determine the convergence or divergence of the series for x =1. (b) Determine the convergence or divergence of the series for x = 1/e. (c) Find the positive values of x for which the series converges.

Calculus: Early Transcendental Functions 6
Riemann Zeta Function The Riemann zeta function for real numbers is defined for all x for which the series \(\zeta(x)=\sum_{n=1}^{\infty} n^{-x}\) converges. Find the domain of the function. Text Transcription: zeta(x)

Calculus: Early Transcendental Functions 6
Review In Exercises 71-82, determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{1}{3 n-2}\) Text Transcription: sum_n=1^infinity 1/3n-2

Calculus: Early Transcendental Functions 6
Review In Exercises 71-82, determine the convergence or divergence of the series. \(\sum_{n=2}^{\infty} \frac{1}{n \sqrt{n^{2}-1}}\) Text Transcription: sum_n=2^infinity 1/n sqrt n^2-1

Calculus: Early Transcendental Functions 6
Review In Exercises 71-82, determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{1}{n \sqrt[4]{n}}\) Text Transcription: sum_n=1^infinity 1/n 4 sqrt n

Calculus: Early Transcendental Functions 6
Review In Exercises 71-82, determine the convergence or divergence of the series. \(3 \sum_{n=1}^{\infty} \frac{1}{n^{0.95}}\) Text Transcription: 3 sum_n=1^infinity n^1/n^0.95

Calculus: Early Transcendental Functions 6
Review In Exercises 71-82, determine the convergence or divergence of the series. \(\sum_{n=0}^{\infty}\left(\frac{2}{3}\right)^{n}\) Text Transcription: sum_n=0^infinity (2/3)^n

Calculus: Early Transcendental Functions 6
Review In Exercises 71-82, determine the convergence or divergence of the series. \(\sum_{n=0}^{\infty}(1.042)^{n}\) Text Transcription: sum_n=0^infinity (1.042)^n

Calculus: Early Transcendental Functions 6
Review In Exercises 71-82, determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^{2}+1}}\) Text Transcription: sum_n=1^infinity n/sqrt n^2+1

Calculus: Early Transcendental Functions 6
Review In Exercises 71-82, determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty}\left(\frac{1}{n^{2}}-\frac{1}{n^{3}}\right)\) Text Transcription: sum_n=1^infinity (1/n^2 - 1/n^3)

Calculus: Early Transcendental Functions 6
Review In Exercises 71-82, determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^{n}\) Text Transcription: sum_n=1^infinity (1+1/n)^n

Calculus: Early Transcendental Functions 6
Review In Exercises 71-82, determine the convergence or divergence of the series. \(\sum_{n=2}^{\infty} \ln n\) Text Transcription: sum_n=2^infinity ln n

Calculus: Early Transcendental Functions 6
Review In Exercises 71-82, determine the convergence or divergence of the series. \(\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{3}}\) Text Transcription: sum_n=2^infinity 1/n(ln n)^3

Calculus: Early Transcendental Functions 6
Review In Exercises 71-82, determine the convergence or divergence of the series. \(\sum_{n=2}^{\infty} \frac{\ln n}{n^{3}}\) Text Transcription: sum_n=2^infinity ln n/n^3