Theory and ExamplesThe Cauchy condensation test The Cauchy

Problem 1E Applying the Integral Test Use the Integral Test to determine if the series in Exercise converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

Problem 3E Applying the Integral Test Use the Integral Test to determine if the series in Exercise converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

Problem 62E Area Repeat Exercise 61, using trapezoids instead of rectangles. That is, on the subinterval (1/(n + 1), 1/n), let denote the area of the trapezoid having heights .y = 1/(n + 1) at x = 1/(n + 1) and y = 1/n at x = 1/n. Reference: Area Consider the sequence On each subinterval (1/(n + 1), 1/n) within the interval [0, 1], erect the rectangle with area having height 1 /n and width equal to the length of the subinterval. Find the total area of all the rectangles. (Hint: Use the result of Example 5 in Section 10.2.)

Problem 4E Applying the Integral Test Use the Integral Test to determine if the series in Exercise converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

Problem 2E Applying the Integral Test Use the Integral Test to determine if the series in Exercise converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

Problem 5E Applying the Integral Test Use the Integral Test to determine if the series in Exercise converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

Problem 6E Applying the Integral Test Use the Integral Test to determine if the series in Exercise converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

Problem 7E Applying the Integral Test Use the Integral Test to determine if the series in Exercise converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

Problem 8E Applying the Integral Test Use the Integral Test to determine if the series in Exercise converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

Problem 10E Applying the Integral Test Use the Integral Test to determine if the series in Exercise converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

Problem 9E Applying the Integral Test Use the Integral Test to determine if the series in Exercise converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

Problem 11E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.) .

Problem 13E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 12E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 18E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 17E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 14E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 19E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 21E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 20E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 23E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 22E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 25E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 24E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 26E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 30E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 28E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 27E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 29E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 31E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 32E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 33E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 34E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 35E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 37E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 38E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 36E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 39E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 40E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

For what values of \(a\), if any, do the series in Exercises 41 and 42 converge? \(\sum_{n=1}^{\infty}\left(\frac{a}{n+2}-\frac{1}{n+4}\right)\) Equation Transcription: Text Transcription: sum_{n=1}^{infty}(frac{a}{n+2}-frac{1}{n+4})

Problem 42E Theory and Examples For what values of a, if any, do the series in exercise converge?

Problem 43E Theory and Examples a. Draw illustrations like those in Figures 10.7 and 10.8 to show that the partial sums of the harmonic series satisfy the inequalities b. There is absolutely no empirical evidence for the divergence of the harmonic series even though we know it diverges. The partial sums just grow too slowly. To see what we mean, suppose you had started with s1 = 1 the day the universe was formed, 13 billion years ago, and added a new term every second. About how large would the partial sum sn be today, assuming a 365-day year? FIGURE 10.7 If the terms of a non-decreasing sequence have an upper bound M, they have a limit L ? M. FIGURE 10.8 As the lengths 1, 1/2, 1/4, 1/8,….., are added one by one, the sum approaches 2.

Problem 47E Theory and Examples diverges a. Use the accompanying graph to show that the partial sum b. What should n be in order that the partial sum

Problem 48E Theory and Examples converges a. Use the accompanying graph to determine the error if s30 = is used to estimate the value of b. Find n so that the partial sum estimates the value of with an error of at most 0.000001.

Problem 44E Theory and Examples Are there any values of x for which converges? Give reasons for your answer

Problem 46E Theory and Examples (Continuation of Exercise 45. ) Is there a “largest” convergent series of positive numbers? Explain. Reference: Exercise 45. Is it true that if is a divergent series of positive numbers, then there is also a divergent series of positive numbers with bn < an for every n? Is there a “smallest” divergent series of positive numbers? Give reasons for your answers.

Problem 49E Theory and Examples Estimate the value of to within 0.01 of its exact value.

Problem 45E Theory and Examples Is it true that if is a divergent series of positive numbers, then there is also a divergent series of positive numbers with bn < an for every n? Is there a “smallest” divergent series of positive numbers? Give reasons for your answers.

Problem 50E Theory and Examples Estimate the value of to within 0.1 of its exact value.

Problem 52E Theory and Examples How many terms of the convergent series should be used to estimate its value with error at most 0.01?

Problem 51E Theory and Examples How many terms of the convergent series should be used to estimate its value with error at most 0.00001?

Problem 54E Theory and Examples Use the Cauchy condensation test from Exercise 53 to show that Reference: Exercise 53 Theory and Examples The Cauchy condensation test The Cauchy condensation test says: Let {an} be a non-increasing sequence ( for all n) of positive terms that converges to 0. Then converges if and only if converges. For example, diverges because diverges. Show why the test works.

Problem 53E Theory and Examples The Cauchy condensation test The Cauchy condensation test says: Let {an} be a non-increasing sequence ( for all n) of positive terms that converges to 0. Then converges if and only if converges. For example, diverges because diverges. Show why the test works.

Problem 56E Theory and Examples (Continuation of Exercise 55.) Use the result in Exercise 55 to determine which of the following series converge and which diverge. Support your answer in each case. Reference: Exercise 55 Logarithmic p -series a. Show that the improper integral converges if and only if p > 1. b. What implications does the fact in part (a) have for the convergence of the series Give reasons for your answer.

Problem 55E Theory and Examples Logarithmic p -series a. Show that the improper integral converges if and only if p > 1. b. What implications does the fact in part (a) have for the convergence of the series Give reasons for your answer.

Euler’s constant Graphs like those in Figure 10.11 suggest that as \(n\) increases there is little change in the difference between the sum \(1+\frac{1}{2}+\cdots+\frac{1}{n}\) and the integral \(\ln n=\int_{1}^{n} \frac{1}{x} d x\) To explore this idea, carry out the following steps. a. By taking \(f(x)=1 / x\) in the proof of Theorem 9, show that \(\ln (n+1) \leq 1+\frac{1}{2}+\cdots+\frac{1}{n} \leq 1+\ln n\) or \(0<\ln (n+1)-\ln n \leq 1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n \leq 1\) Thus, the sequence \(a_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n\) is bounded from below and from above. b. Show that \(\frac{1}{n+1}<\int_{n}^{n+1} \frac{1}{x} d x=\ln (n+1)-\ln n\) and use this result to show that the sequence \(\left\{a_{n}\right\}\) in part (a) is decreasing. Since a decreasing sequence that is bounded from below converges, the numbers \(a_{n}\) defined in part (a) converge: \(1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n \rightarrow \gamma\) The number \(\gamma\) whose value is 0.5772 . . ., is called Euler’s constant. Equation Transcription: Text Transcription: n 1+1/2+cdots+1/n ln n= integral over 1 ^n 1/x dx f(x)=1/x ln(n+1)leq 1+1/2+cdots+ 1/n leq 1+ln n 0<ln(n+1)-ln n1+1/2+cdots+1/n-ln n leq 1 a_n=1+1/2+cdots+1/n-ln n 1/n + 1 < integral over n ^n+1 1/x dx=ln (n+1)-ln n {a_n} a_n 1+1/2+cdots+1/n-ln n rightarrow gamma gamma

Problem 58E Theory and Examples Use the Integral Test to show that the series converges.

Problem 59E Theory and Examples a. For the series , use the inequalities in Equation (2) with n = 10 to find an interval containing the sum S. b. As in Example 5, use the midpoint of the interval found in part (a) to approximate the sum of the series. What is the maximum error for your approximation? Reference: Example 5 EXAMPLE 5 Estimate the sum of the series using the inequalities in (2) and n = 10

Problem 60E Theory and Examples Repeat Exercise 59 using the series Reference: Exercise 59 a. For the series , use the inequalities in Equation (2) with n = 10 to find an interval containing the sum S. b. As in Example 5, use the midpoint of the interval found in part (a) to approximate the sum of the series. What is the maximum error for your approximation? EXAMPLE 5 Estimate the sum of the series using the inequalities in (2) and n = 10

Problem 61E Area Consider the sequence On each subinterval (1/(n + 1), 1/n) within the interval [0, 1], erect the rectangle with area having height 1 /n and width equal to the length of the subinterval. Find the total area of all the rectangles. (Hint: Use the result of Example 5 in Section 10.2.)

Problem 15E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

Problem 16E Determining Convergence or Divergence Which of the series in Exercise converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)