Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ 2/(5k – 1)
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Determine the third and fourth Taylor polynomials of x3 + 3x - 1 at x = -1.
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1/(1 – x), ex, or cos x. These series are derived in Examples 1 and 2 and Check Your Understanding Probl
> Find the Taylor series at x = 0 of the given function by computing three or four derivatives and using the definition of the Taylor series. (1 + x)3
> Find the Taylor series at x = 0 of the given function by computing three or four derivatives and using the definition of the Taylor series. √(1 + x)
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ 5/k3/2
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ 3/√k
> Determine the third and fourth Taylor polynomials of cos x at x = π.
> Use Exercise 30 to show that the series ∑k=1∞ 3/k2 is convergent. Then, use the comparison test to show that the series ∑k=1∞ e1/k/k2 is convergent. Exercise 30: Let ∑k=1∞ ak be a convergent series with sum S, and let c be a constant. Then, ∑k=2∞ cak is
> Use Exercise 29 to show that the series ∑k=0∞ (8k + 9k)/10k is convergent, and determine its sum. Exercise 29: The following property is true for any two series (with possibly some negative terms): Let ∑k=1∞ ak and ∑k=1∞ bk be convergent series whose su
> Let ∑k=1∞ ak be a convergent series with sum S, and let c be a constant. Then, ∑k=2∞ cak is a convergent series whose sum is c * S. Make a geometric picture to illustrate why this is true when c = 2 and the terms ak are all positive.
> The following property is true for any two series (with possibly some negative terms): Let ∑k=1∞ ak and ∑k=1∞ bk be convergent series whose sums are S and T, respectively. Then, ∑k=1∞ (ak + bk) is a convergent series whose sum is S + T. Make a geometric
> Can the comparison test be used with a ∑k=1∞ 1/(k2 ln k) and ∑k=2∞ 1/k2 to deduce anything about the first series?
> Can the comparison test be used with ∑k=2∞ 1/(k ln k) and ∑k=2∞ 1/k to deduce anything about the first series?
> Use the comparison test to determine whether the infinite series is convergent or divergent. ∑k=0∞ 1/(3/4)k + (5/4)k [Compare with ∑k=0∞ (3/4)-k or ∑k=0∞ (5/4)-k.]
> Use the comparison test to determine whether the infinite series is convergent or divergent. ∑k=1∞ 1/5k cos2 (kπ/4) [Compare with ∑k=1∞ 1/5k.]
> Use the comparison test to determine whether the infinite series is convergent or divergent. ∑k=1∞ 1/k3k [Compare with ∑k=1∞ 1/3k.]
> Use the comparison test to determine whether the infinite series is convergent or divergent. ∑k=1∞ 1/(2k + k) [Compare with ∑k=1∞ 1/2k.]
> Determine the fourth Taylor polynomial of ln x at x = 1.
> Use the comparison test to determine whether the infinite series is convergent or divergent. ∑k=2∞ 1/√(k2 – 1) [Compare with ∑k=2∞ 1/k.]
> Use the comparison test to determine whether the infinite series is convergent or divergent. ∑k=2∞ 1/(k2 + 5) [Compare with ∑k=2∞ 1/k2.]
> Is the series a ∑k=1∞ 3k/4k convergent? What is the easiest way to answer this question? Can you tell if ∫1∞3x/4x dx is convergent?
> It can be shown that lim b → ∞ be-b = 0. Use this fact and the integral test to show that a ∑k=1∞ k ek is convergent.
> Use the integral test to determine if a ∑k=1∞ e1/k k2 is convergent. Show that the hypotheses of the integral test are satisfied.
> It can be shown that ∫0∞ 3/(9 + x2) dx is convergent. Use this fact to show that an appropriate infinite series converges. Give the series, and show that the hypotheses of the integral test are satisfied.
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=2∞ (k + 1)/(k2 + 2k + 1)2
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ (2k + 1)/(k2 + k + 2)
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ k-3/4
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ ke-k2
> Determine the third Taylor polynomial of 1/(5 – x) at x = 4.
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ 1/e2k+1
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ e3-k
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ 1/(3k)2
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=2∞ 1/k(ln k)2
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=1∞ 1/(2k + 1)3
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=2∞ k/(k2 + 1)3/2
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=2∞ 1/k√(ln k)
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=0∞ 7/(k + 100)
> Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied.) ∑k=2 ∞ 1/(k - 1)3
> Use a second Taylor polynomial at x = 0 to estimate the area under the curve y = √(cos x) from x = -1 to x = 1. (The exact answer to three decimal places is 1.828.)
> Determine the sums of the following geometric series when they are convergent. 1 + 1/6 + 1/62 + 1/63 + 1/64 …
> Convince yourself that the equation is correct by summing up the first 999 terms of the infinite series and comparing the sum with the value on the right. ∑x=1 ∞ (-1)x+1 / x = ln 2
> Convince yourself that the equation is correct by summing up the first 999 terms of the infinite series and comparing the sum with the value on the right. ∑x=1 ∞ 1/x2 = π2/6
> The sum of the first n odd numbers is n2; that is, ∑x=1 n (2x - 1) = n2. Verify this formula for n = 5, 10, and 25.
> Verify the formula ∑ x=1 n x = n(n + 1) / 2 for n = 10, 50, and 100.
> The calculator screen computes a partial sum of an infinite series. Write out the first five terms of the series and determine the exact value of the infinite series.
> The calculator screen computes a partial sum of an infinite series. Write out the first five terms of the series and determine the exact value of the infinite series.
> What is the exact value of the infinite geometric series whose partial sum appears at the second entry in Fig. 2? Figure 2:
> What is the exact value of the infinite geometric series whose partial sum appears at the first entry in Fig. 3? Figure 3:
> Show that the infinite series 1 + ½ + 1/3 + ¼ + 1/5 + … diverges.
> Use a second Taylor polynomial at x = 0 to estimate the area under the curve y = ln(1 + x2) from x = 0 to x = 1/2.
> Determine the third Taylor polynomial of the given function at x = 0. f (x) = 1/(x + 2)
> What is routing?
> How does TCP/IP perform address resolution from IP addresses into data link layer addresses?
> How does TCP/IP perform address resolution from URLs into network layer addresses?
> What is address resolution?
> What benefits and problems does dynamic addressing provide?
> What does the transport layer do?
> Briefly define noise.
> Is there any difference in the error rates of lower-speed lines and higher-speed lines?
> Errors normally appear in _____, which is when more than 1 data bit is changed by the error-causing condition.
> Define two fundamental types of errors.
> What are the benefits of cloud computing?
> Which is better, controlled access or contention? Explain.
> Compare and contrast roll-call polling, hub polling (or token passing), and contention.
> Under what conditions is media access control unimportant?
> Show how the word “HI” would be sent using asynchronous transmission using even parity (make assumptions about the bit patterns needed). Show how it would be sent using Ethernet.
> What media access control technique does your class use?
> Are large frame sizes better than small frame sizes? Explain.
> Under what conditions does a data link layer protocol need an address?
> Are stop bits necessary in asynchronous transmission? Explain by using a diagram.
> How do information bits differ from overhead bits?
> What is transmission efficiency?
> How does a thin client differ from a thick client?
> Describe the frame layouts for SDLC, Ethernet, and PPP.
> Which is the simplest (least sophisticated) protocol described in this chapter?
> What is media access control, and why is it important?
> Briefly describe how continuous ARQ works.
> Under what circumstances is forward error correction desirable?
> How does forward error-correction work? How is it different from other error-correction methods?
> How does CRC work?
> Briefly describe how checksum works.
> Briefly describe how even parity and odd parity work.
> Describe three approaches to detecting errors, including how they work, the probability of detecting an error, and any other benefits or limitations.
> Compare and contrast two-tier, three-tier, and n-tier client–server architectures. What are the technical differences, and what advantages and disadvantages does each offer?
> What are the three ways of reducing errors and the types of noise they affect?
> How do amplifiers differ from repeaters?
> Describe four types of noise. Which is likely to pose the greatest problem to network managers?
