Chapter 10

Determine whether the given geometric series converges or diverges. If the series converges, find its sum. $$ \sum_{n=0}^{\infty} \frac{(-2)^{n}}{5^{n+1}} $$

Determine whether the given geometric series converges or diverges. If the series converges, find its sum. $$ \sum_{n=1}^{\infty} \frac{4^{n-1}}{3^{2 n}} $$

Determine whether the given geometric series converges or diverges. If the series converges, find its sum. $$ \sum_{n=0}^{\infty} \frac{13}{(-5)^{n}} $$

Determine whether the given geometric series converges or diverges. If the series converges, find its sum. $$ \sum_{n=0}^{\infty}\left(-\frac{3}{2}\right)^{n} $$

Determine whether the given geometric series converges or diverges. If the series converges, find its sum. $$ \sum_{n=0}^{\infty} e^{-0.5 n} $$

Determine whether the given geometric series converges or diverges. If the series converges, find its sum. $$ \sum_{n=1}^{\infty} \frac{2^{n+1}}{3^{n-1}} $$

Determine whether the given geometric series converges or diverges. If the series converges, find its sum. $$ \sum_{n=0}^{\infty}\left[\left(\frac{2}{3}\right)^{n}+\left(\frac{3}{2}\right)^{n}\right] $$

Determine whether the given geometric series converges or diverges. If the series converges, find its sum. $$ \sum_{n=0}^{\infty}\left(\frac{3}{2}\right)\left(\frac{2}{3}\right)^{n} $$

Determine whether the given series converges or diverges. $$ \sum_{n=0}^{\infty} \frac{1}{2 n+1} $$

Determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^{2} $$

Determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \ln \left(2+\frac{1}{n}\right) $$

Determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{1}{n^{2} \sqrt{n}} $$

Determine whether the given series converges or diverges. $$ \sum_{n=2}^{\infty} \frac{\ln \sqrt{n}}{\sqrt{n}} $$

Determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{10^{n}}{n !} $$

Determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{n^{3}}{3^{n}} $$

Determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{(-2)^{n}}{n^{2}} $$

Determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{(-3)^{2 n}}{n !} $$

Determine whether the given series converges or diverges. $$ \sum_{n=2}^{\infty} \frac{1}{n^{2}-1} $$

Find the interval of absolute convergence for the given power series. $$ \sum_{n=0}^{\infty}(3 x)^{n} $$

Find the interval of absolute convergence for the given power series. $$ \sum_{n=1}^{\infty} \frac{(-2 x)^{n}}{n !} $$

Find the Taylor series at \(x=0\) for the given function, either by using the definition or by manipulating a known series. $$ f(x)=e^{x}-e^{-x} $$

Find the Taylor series at \(x=0\) for the given function, either by using the definition or by manipulating a known series. $$ f(x)=\frac{1}{(1+2 x)^{2}} $$

Find the Taylor series at \(x=0\) for the given function, either by using the definition or by manipulating a known series. $$ f(x)=\ln \left(\frac{x+1}{2 x+1}\right) $$

Find the Taylor series at \(x=0\) for the given function, either by using the definition or by manipulating a known series. $$ f(x)=x^{2} e^{-2 x} $$

Find the Taylor series for the given function at the specified value of \(x=a\). $$ f(x)=\ln (2+x) ; a=-1 $$

Find the Taylor series for the given function at the specified value of \(x=a\). $$ f(x)=x \ln x ; a=1 $$

Find the Taylor series for the given function at the specified value of \(x=a\). $$ f(x)=\frac{1-x}{1+x} ; a=0 $$

Calculate the first four Taylor polynomials \(P_{0}(x)\), \(P_{1}(x), P_{2}(x)\), and \(P_{3}(x)\) at \(x=0\) for the function $$ f(x)=\frac{1}{\sqrt{1-x^{2}}} $$ a. Use the graphing utility on your calculator to sketch the graphs of these four polynomials on the same set of axes. b. Use \(P_{3}(x)\) to estimate the value of the definite integral $$ \int_{0}^{1 / 2} \frac{d x}{\sqrt{1-x^{2}}} $$

Use your calculator to compute the sum \(S(N)=1-\frac{1}{2 !}+\frac{1}{4 !}-\frac{1}{6 !}+\cdots+\frac{(-1)^{N}}{(2 N) !}\) for \(N=2,7,10,15\), and 30 . Based on your results, do you think the series $$ \sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2 k) !} $$ converges or diverges? If it converges, what do you think its sum will be?

A ball is dropped from a height of \(H\) feet and bounces indefinitely, repeatedly rebounding to \(75 \%\) of its previous height. If it travels a total distance of 70 feet, what is \(H\) ?

An investment guarantees annual payments of \(\$ 5,000\) in perpetuity, with the payments beginning immediately. Find the present value of this investment if the prevailing annual interest rate remains fixed at \(5 \%\) compounded continuously.

Suppose that nationwide, approximately \(91 \%\) of all income is spent and \(9 \%\) is saved. What is the total amount of spending generated by a 60 billion dollar tax rebate if saving habits do not change?

A patient is given an injection of 25 units of a certain drug every 24 hours. The drug is eliminated exponentially so that the fraction that remains in the patient's body after \(t\) days is \(f(t)=e^{-k t}\) for some constant \(k\). If 70 units of drug eventually accumulate in the patient's bloodstream just prior to an injection, what is \(k\) ?

A developing country currently has 2,500 trained scientists. The government estimates that each year, \(6 \%\) of the current number of scientists either retire, die, or emigrate, while 278 new scientists graduate from college. If these trends continue, how many scientists will there be in 20 years? How many in the long run?

In a classic genetic model," the average life span of a harmful gene is related to the infinite series $$ 1+2 r+3 r^{2}+4 r^{3}+\cdots=\sum_{k=1}^{\infty} k r^{k-1} $$ for \(0

Linguists and psychologists who are interested in the evolution of language have noticed an interesting pattern in the frequency of so- called rare words in certain literary works. According to one classic model, \({ }^{+}\)if a book contains a total of \(T\) different such words, then approximately \(\frac{T}{(1)(2)}\) words appear exactly once, \(\frac{T}{(2)(3)}\) words appear exactly twice, and in general \(\frac{T}{k(k+1)}\) words appear exactly \(k\) times. a. Assuming the word frequency pattern in the model is accurate, why should you expect the series $$ \sum_{k=1}^{\infty} \frac{T}{k(k+1)} $$ to converge? What would you expect its sum to be? b. Verify your conjecture in part (a) by actually summing the series. [Hint: See Example 10.1.3.] c. Read an article on information and learning theory, and write a paragraph on mathematical methods in this subject.

Economists refer to the process of moving from one job to another as labor migration. Such movement is usually undertaken as a means for social or economic improvement, but it also involves costs, such as the loss of seniority in the old job and the psychological cost of disrupting relationships. Consider the function \(^{+}\) $$ V=\sum_{n=1}^{N} \frac{E_{2}(n)-E_{1}(n)}{(1+i)^{n}}-\sum_{n=1}^{N} \frac{C_{m}(n)}{(1+i)^{n}}-C_{p} $$ \({ }^{*} \mathrm{C} . \mathrm{C} . \mathrm{Li}\), Human Genetics: Principles and Methods, New York: McGraw-Hill. 'G. K. Zipf, Human Behavior and the Principle of Least Effort, Cambridge, MA: Addison-Wesley. Another good source is C. E. Shannon and W. Weaver, The Mathematical Theory of Communication, Urbana, IL: Univ. of Illinois Press. \({ }^{2} \mathrm{C}\). R. McConnell and S. L. Brue, Contemporary Labor Economics, New York: MoGraw-Hill, 1992, pp. 440-444. where \(E_{2}(n)\) and \(E_{1}(n)\) denote the earnings from the new and old jobs, respectively, in year \(n\) after the move is to be made; \(i\) is the prevailing annual interest rate (in decimal form); \(N\) is the number of years the person is expected to be on the new job; and \(C_{m}\) and \(C_{p}\) are the expected monetary and net psychological costs of the move (psychological gain minus psychological loss). a. What does \(V\) represent? Why is the job move desirable if \(V>0\) and undesirable if \(V<0\) ? b. For simplicity, assume that \(E_{2}-E_{1}\) and \(C_{m}\) are constant for all \(n\) and that the person expects to stay on the new job "forever" once he or she moves (that is, \(N \rightarrow \infty\) ). Find a formula for \(V\), and use it to obtain a criterion for whether or not the job move should be made. [Hint: Your criterion should be an inequality involving \(E_{1}, E_{2}, C_{m}, C_{p}\), and \(i\).] c. Read an article on job mobility and labor migration, and write a paragraph on mathematical methods for modeling such issues.

At the time we studied carbon dating in Chapter 4 , we noted that radiocarbon methods are used mainly for dating specimens that are not too old.* To date rocks or artifacts that are older than 40,000 years, it is necessary to use other methods. Whether \({ }^{14} \mathrm{C}\) or a radioactive isotope of some other element is used for dating, it can be shown that $$ \frac{S(t)-S(0)}{R(t)}+1=e^{(\ln 2) t / \lambda} $$ where \(R(t)\) is the number of atoms of radioactive isotope at time \(t, S(t)\) is the number of atoms of the stable product of radioactive decay, \(S(0)\) is the number of atoms of stable product initially present (at \(t=0\) ), and \(\lambda\) is the half-life of the radioactive isotope (the time it takes for half a sample to decay). a. Approximate the time \(t\) in this formula using the Taylor polynomial of degree 2 for \(e^{x}\) at \(x=0\). b. Suppose a piece of mica is analyzed, and it is found that \(5 \%\) of the atoms in the rock are radioactive rubidium- 87 and \(0.04 \%\) are strontium-87. If all the strontium-87 was produced by decay of the rubidium \(-87\) in the rock, how old is the rock? Use the approximation obtained in part (a). You will

The story goes that the famous mathematician, John von Neumann (1903-1957), was once challenged to solve a version of the following problem: Two trains, each traveling at \(30 \mathrm{ft} / \mathrm{sec}\), approach each other on a straight track. When they are 1,000 feet apart, a bee begins flying from one train to the other and back again at the rate of \(60 \mathrm{ft} / \mathrm{sec}\), and continues to do so until the trains crash. How far does the bee fly before it is crushed by the crashing trains? Von Neumann pondered the question only briefly before giving the correct answer. The poser of the problem chuckled appreciatively and said, "You saw the trick. I should have known better than to try to fool you, Professor." Von Neumann looked puzzled. "What trick?" he replied, "I summed the series." a. Sum a series as von Neumann did to find the distance traveled by the ill- fated bee. b. Unlike von Neumann, do you see an easy way to solve this problem? (Incidentally, a form of this question is sometimes used by Microsoft to test new employees.) c. Try this kinder, gentler version of the same problem. Suppose the conductors of the two trains see each other and hit the brakes when they are 180 feet apart. If both trains decelerate at the rate of \(5 \mathrm{ft} / \mathrm{sec}^{2}\), how far does the bee fly before the trains come together?