MATHEMATICAL METHODS IN THE PHY SCIENC
3rd Edition
ISBN: 9781118048870
Author: Boas
Publisher: WILEY
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Textbook Question
Chapter 1.15, Problem 18P
Find the sum of each of the following series by recognizing it the Maclaurin series for a function evaluated at a point.
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Calculus lll
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For each month of the year, Taylor collected the average high temperatures in Jackson, Mississippi. He used the data to create the histogram shown. Which set of data did he use to create the histogram?
A
55, 60, 64, 72, 73, 75, 77, 81, 83, 91, 91, 92\ 55,\ 60,\ 64,\ 72,\ 73,\ 75,\ 77,\ 81,\ 83,\ 91,\ 91,\ 92 55, 60, 64, 72, 73, 75, 77, 81, 83, 91, 91, 92
B
55, 57, 60, 65, 70, 71, 78, 79, 85, 86, 88, 91\ 55,\ 57,\ 60,\ 65,\ 70,\ 71,\ 78,\ 79,\ 85,\ 86,\ 88,\ 91 55, 57, 60, 65, 70, 71, 78, 79, 85, 86, 88, 91
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55, 60, 63, 64, 65, 71, 83, 87, 88, 88, 89, 93\ 55,\ 60,\ 63,\ 64,\ 65,\ 71,\ 83,\ 87,\ 88,\ 88,\ 89,\ 93 55, 60, 63, 64, 65, 71, 83, 87, 88, 88, 89, 93
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In this problem, we consider a Brownian motion (W+) t≥0. We consider a stock model (St)t>0
given (under the measure P) by
d.St 0.03 St dt + 0.2 St dwt,
with So 2. We assume that the interest rate is r = 0.06. The purpose of this problem is to
price an option on this stock (which we name cubic put). This option is European-type, with
maturity 3 months (i.e. T = 0.25 years), and payoff given by
F = (8-5)+
(a) Write the Stochastic Differential Equation satisfied by (St) under the risk-neutral measure
Q. (You don't need to prove it, simply give the answer.)
(b) Give the price of a regular European put on (St) with maturity 3 months and strike K = 2.
(c) Let X =
S. Find the Stochastic Differential Equation satisfied by the process (Xt)
under the measure Q.
(d) Find an explicit expression for X₁ = S3 under measure Q.
(e) Using the results above, find the price of the cubic put option mentioned above.
(f) Is the price in (e) the same as in question (b)? (Explain why.)
Chapter 1 Solutions
MATHEMATICAL METHODS IN THE PHY SCIENC
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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