Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
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Textbook Question
Chapter 1.6, Problem 27P
Use the ratio test to find whether the following series converge or diverge:
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8.1.6The yield of a chemical process is being studied. From previous experience, yield is known to be normally
distributed and σ = 3. The past 5 days of plant operation have
resulted in the following percent yields: 91.6, 88.75, 90.8, 89.95,
and 91.3. Find a 95% two-sided confidence interval on the true
mean yield.
8.1.7 .A manufacturer produces piston rings for an automobile engine. It is known that ring diameter is normally distributed
with σ = 0.001 millimeters. A random sample of 15 rings has a
mean diameter of x = 74.036 millimeters.
a. Construct a 99% two-sided confidence interval on the
mean piston ring diameter.
b. Construct a 99% lower-confidence bound on the mean
piston ring diameter. Compare the lower bound of this confi-
dence interval with the one in part (a).
8.1.2 .Consider the one-sided confidence interval expressions for a mean of a normal population.
a. What value of zα would result in a 90% CI?
b. What value of zα would result in a 95% CI?
c. What value of zα would result in a 99% CI?
8.1.3 A random sample has been taken from a normal distribution and the following confidence intervals constructed using the
same data: (38.02, 61.98) and (39.95, 60.05)
a. What is the value of the sample mean?
b. One of these intervals is a 95% CI and the other is a
90% CI. Which one is the 95% CI and why?
8.1.4 . A confidence interval estimate is desired for the gain
in a circuit on a semiconductor device. Assume that gain is normally distributed with standard deviation σ = 20.
a. How large must n be if the length of the 95% CI is to
be 40?
b. How large must n be if the length of the 99% CI is to
be 40?
8.1.5 Suppose that n = 100 random samples of water from
a freshwater lake were taken and the calcium concentration
(milligrams per liter) measured. A 95% CI on the mean calcium
concentration is 0.49 g μ g 0.82.
a. Would a 99% CI calculated from the same sample data be
longer or shorter?
b. Consider the following statement: There is a 95% chance
that μ is between 0.49 and 0.82. Is this statement correct?
Explain your answer.
c. Consider the following statement: If n = 100 random
samples of water from the lake were taken and the 95% CI on
μ computed, and this process were repeated 1000 times, 950
of the CIs would contain the true value of μ. Is this statement
correct? Explain your answer
Chapter 1 Solutions
Mathematical Methods in the Physical Sciences
Ch. 1.1 - In the bouncing ball example above, find the...Ch. 1.1 - Derive the formula (1.4) for the sum Sn of the...Ch. 1.1 - Use equation (1.8) to find the fractions that are...Ch. 1.1 - Use equation (1.8) to find the fractions that are...Ch. 1.1 - Use equation (1.8) to find the fractions that are...Ch. 1.1 - Use equation (1.8) to find the fractions that are...Ch. 1.1 - Use equation (1.8) to find the fractions that are...Ch. 1.1 - Use equation (1.8) to find the fractions that are...Ch. 1.1 - Use equation (1.8) to find the fractions that are...Ch. 1.1 - Use equation (1.8) to find the fractions that are...
Ch. 1.1 - Use equation (1.8) to find the fractions that are...Ch. 1.1 - In a water purification process, one-nth of the...Ch. 1.1 - If you invest a dollar at 6% interest compounded...Ch. 1.1 - A computer program gives the result 1/6 for the...Ch. 1.1 - Connect the midpoints of the sides of an...Ch. 1.1 - Suppose a large number of particles are bouncing...Ch. 1.2 - In the following problems, find the limit of the...Ch. 1.2 - In the following problems, find the limit of the...Ch. 1.2 - In the following problems, find the limit of the...Ch. 1.2 - In the following problems, find the limit of the...Ch. 1.2 - In the following problems, find the limit of the...Ch. 1.2 - In the following problems, find the limit of the...Ch. 1.2 - In the following problems, find the limit of the...Ch. 1.2 - In the following problems, find the limit of the...Ch. 1.2 - In the following problems, find the limit of the...Ch. 1.4 - For the following series, write formulas for the...Ch. 1.4 - For the following, write formulas for the...Ch. 1.4 - For the following series, write formulas for the...Ch. 1.4 - For the following series, write formulas for the...Ch. 1.4 - For the following series, write formulas for the...Ch. 1.4 - For the following series, write formulas for the...Ch. 1.4 - For the following series, write formulas for the...Ch. 1.5 - Use the preliminary test to decide whether the...Ch. 1.5 - Use the preliminary test to decide whether the...Ch. 1.5 - Use the preliminary test to decide whether the...Ch. 1.5 - Use the preliminary test to decide whether the...Ch. 1.5 - Use the preliminary test to decide whether the...Ch. 1.5 - Use the preliminary test to decide whether the...Ch. 1.5 - Use the preliminary test to decide whether the...Ch. 1.5 - Use the preliminary test to decide whether the...Ch. 1.5 - Use the preliminary test to decide whether the...Ch. 1.5 - Use the preliminary test to decide whether the...Ch. 1.5 - Using (4.6), give a proof of the preliminary test....Ch. 1.6 - Show that n! 2 for all n 3. Hint: Write out a...Ch. 1.6 - Prove that the harmonic series n=11/n is divergent...Ch. 1.6 - Prove the convergence n=11/n2 by grouping terms...Ch. 1.6 - Use the comparison test to prove the convergence...Ch. 1.6 - Test the following series for convergence using...Ch. 1.6 - There are 9 one-digit numbers (1 to 9), 90...Ch. 1.6 - Use the integral test to find whether the...Ch. 1.6 - Use the integral test to find whether the...Ch. 1.6 - Use the integral test to find whether the...Ch. 1.6 - Use the integral test to find whether the...Ch. 1.6 - Use the integral test to find whether the...Ch. 1.6 - Use the integral test to find whether the...Ch. 1.6 - Use the integral test to find whether the...Ch. 1.6 - Use the integral test to find whether the...Ch. 1.6 - Use the integral test to prove the following...Ch. 1.6 - In testing 1/n2 for convergence, a student...Ch. 1.6 - Use the integral test to show that n=0en2...Ch. 1.6 - Use the ratio test to find whether the following...Ch. 1.6 - Use the ratio test to find whether the following...Ch. 1.6 - Use the ratio test to find whether the following...Ch. 1.6 - Use the ratio test to find whether the following...Ch. 1.6 - Use the ratio test to find whether the following...Ch. 1.6 - Use the ratio test to find whether the following...Ch. 1.6 - Use the ratio test to find whether the following...Ch. 1.6 - Use the ratio test to find whether the following...Ch. 1.6 - Use the ratio test to find whether the following...Ch. 1.6 - Use the ratio test to find whether the following...Ch. 1.6 - Use the ratio test to find whether the following...Ch. 1.6 - Use the ratio test to find whether the following...Ch. 1.6 - Prove the ratio test. Hint: If an+1/an1, take ...Ch. 1.6 - Use the special comparison test to find whether...Ch. 1.6 - Use the special comparison test to find whether...Ch. 1.6 - Use the special comparison test to find whether...Ch. 1.6 - Use the special comparison test to find whether...Ch. 1.6 - Use the special comparison test to find whether...Ch. 1.6 - Use the special comparison test to find whether...Ch. 1.6 - Prove the special comparison test. Hint (part a):...Ch. 1.7 - Test the following series for convergence....Ch. 1.7 - Test the following series for convergence....Ch. 1.7 - Test the following series for convergence....Ch. 1.7 - Test the following series for convergence....Ch. 1.7 - Test the following series for convergence....Ch. 1.7 - Test the following series for convergence....Ch. 1.7 - Test the following series for convergence....Ch. 1.7 - Test the following series for convergence....Ch. 1.7 - Prove that an absolutely convergent series n=1an...Ch. 1.7 - The following alternating series are divergent...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.9 - Test the following series for convergence or...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - Find the interval of convergence of each of the...Ch. 1.10 - The following series are not power series, but you...Ch. 1.10 - The following series are not power series, but you...Ch. 1.10 - The following series are not power series, but you...Ch. 1.10 - The following series are not power series, but you...Ch. 1.10 - The following series are not power series, but you...Ch. 1.10 - The following series are not power series, but you...Ch. 1.10 - The following series are not power series, but you...Ch. 1.12 - By the method used to obtain (12.5) [which is the...Ch. 1.13 - Use the ratio test to show that a binomial series...Ch. 1.13 - Show that the binomial coefficients 1n=(1)n.Ch. 1.13 - Show that if p is a positive integer, then pn=0...Ch. 1.13 - Write the Maclaurin series for 1/1+x in form...Ch. 1.13 - Using the methods of this section: Find the first...Ch. 1.13 - Using the methods of this section: Find the first...Ch. 1.13 - Using the methods of this section: Find the first...Ch. 1.13 - Using the methods of this section: Find the first...Ch. 1.13 - Using the methods of this section: Find the first...Ch. 1.13 - Using the methods of this section: Find the first...Ch. 1.13 - Using the methods of this section: Find the first...Ch. 1.13 - Using the methods of this section: Find the first...Ch. 1.13 - Using the methods of this section: Find the first...Ch. 1.13 - Using the methods of this section: Find the first...Ch. 1.13 - Using the methods of this section: Find the first...Ch. 1.13 - Using the methods of this section: Find the first...Ch. 1.13 - Using the methods of this section: Find the first...Ch. 1.13 - Using the methods of this section: Find the first...Ch. 1.13 - Using the methods of this section: Find the first...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - In cos x Hints: Method l: Write cos x = 1+(cos...Ch. 1.13 - Find the first few terms of the Maclaurin series...Ch. 1.13 - Using method F above, find the first few terms of...Ch. 1.13 - Using method F above, find the first few terms of...Ch. 1.13 - Using method F above, find the first few terms of...Ch. 1.13 - Using method F above, find the first few terms of...Ch. 1.13 - Using method F above, find the first few terms of...Ch. 1.13 - Using method F above, find the first few terms of...Ch. 1.14 - Prove theorem (14.3). Hint: Group the terms in the...Ch. 1.14 - Using computer or tables (or Chapter 7, Section...Ch. 1.14 - In Problem 3 to 7, assume that the Maclaurin...Ch. 1.14 - In Problem 3 to 7, assume that the Maclaurin...Ch. 1.14 - In Problem 3 to 7, assume that the Maclaurin...Ch. 1.14 - In Problem 3 to 7, assume that the Maclaurin...Ch. 1.14 - In Problem 3 to 7, assume that the Maclaurin...Ch. 1.14 - Estimate the error if n=1xn/n3 is approximated by...Ch. 1.14 - Consider the series in Problem 4.6 and show that...Ch. 1.14 - Show that the interval of convergence of the...Ch. 1.14 - Show that the Maclaurin series for sin x converges...Ch. 1.14 - Show as in Problem 11 that the Maclaurin series...Ch. 1.14 - Show that Maclaurin for (1+x)p converges to (1+x)p...Ch. 1.15 - In problems 1 to 4, use power series to evaluate...Ch. 1.15 - In problems 1 to 4, use power series to evaluate...Ch. 1.15 - In problems 1 to 4, use power series to evaluate...Ch. 1.15 - In problems 1 to 4, use power series to evaluate...Ch. 1.15 - Use Maclaurin series to evaluate each of the...Ch. 1.15 - Use Maclaurin series to evaluate each of the...Ch. 1.15 - Use Maclaurin series to evaluate each of the...Ch. 1.15 - Use Maclaurin series to evaluate each of the...Ch. 1.15 - Use Maclaurin series to evaluate each of the...Ch. 1.15 - Use Maclaurin series to evaluate each of the...Ch. 1.15 - Use Maclaurin series to evaluate each of the...Ch. 1.15 - Use Maclaurin series to evaluate each of the...Ch. 1.15 - Use Maclaurin series to evaluate each of the...Ch. 1.15 - Find a two term aproximation for each of the...Ch. 1.15 - Find a two term aproximation for each of the...Ch. 1.15 - Find the sum of each of the following series by...Ch. 1.15 - Find the sum of each of the following series by...Ch. 1.15 - Find the sum of each of the following series by...Ch. 1.15 - Find the sum of each of the following series by...Ch. 1.15 - By computer or tables, find the exact sum of each...Ch. 1.15 - By computer, find a numerical approximation for...Ch. 1.15 - The series n=11/n8,s1, is called the Riemann Zeta...Ch. 1.15 - Find the following limits using Maclaurin series...Ch. 1.15 - Evaluate the following indeterminate forms by...Ch. 1.15 - In general, we do not expect Maclaurin series to...Ch. 1.15 - Find the values of several derivatives of...Ch. 1.15 - The velocity of electrons from a high energy...Ch. 1.15 - The energy of an electron at speed in special...Ch. 1.15 - The figure shows a heavy weight suspended by a...Ch. 1.15 - Prob. 30PCh. 1.15 - A tall tower of circular cross section is...Ch. 1.15 - Show that the doubling time (time for your money...Ch. 1.15 - If you are at the top Of a tower Of height h above...Ch. 1.16 - Show that it is possible to stack a pile of...Ch. 1.16 - The picture is a mobile constructed of dowels (or...Ch. 1.16 - Show that n=21/n3/2 is convergent. What is wrong...Ch. 1.16 - Test for convergence: n=12nn!Ch. 1.16 - Test for convergence: n=2(n1)21+n2Ch. 1.16 - Test for convergence: n=2n1(n+1)21Ch. 1.16 - Test for convergence: n=21n1n(n)3Ch. 1.16 - Test for convergence: n=22n3n42Ch. 1.16 - Find the interval of convergence, including...Ch. 1.16 - Find the interval of convergence, including...Ch. 1.16 - Find the interval of convergence, including...Ch. 1.16 - Find the interval of convergence, including...Ch. 1.16 - Find the interval of convergence, including...Ch. 1.16 - Find the Maclaurin series for the folliwing...Ch. 1.16 - Find the Maclaurin series for the folliwing...Ch. 1.16 - Find the Maclaurin series for the folliwing...Ch. 1.16 - Find the Maclaurin series for the folliwing...Ch. 1.16 - Find the Maclaurin series for the folliwing...Ch. 1.16 - Find the few terms of the Taylor series for the...Ch. 1.16 - Find the few terms of the Taylor series for the...Ch. 1.16 - Find the few terms of the Taylor series for the...Ch. 1.16 - Use the series you know to show that:...Ch. 1.16 - Use the series you know to show that:...Ch. 1.16 - Use the series you know to show that:...Ch. 1.16 - Evaluate the limit limx0x2/1ncosx by series (in...Ch. 1.16 - Use Maclaurin to do Problem 26 to 29 and check...Ch. 1.16 - Use Maclaurin to do Problem 26 to 29 and check...Ch. 1.16 - Use Maclaurin to do Problem 26 to 29 and check...Ch. 1.16 - Use Maclaurin to do Problem 26 to 29 and check...Ch. 1.16 - It is clear that you (or your computer) cant find...Ch. 1.16 - As in Problem 30, for each of the following...
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