Chapter 18

Find the sum of the following. (If there is no nite sum, say so.) (a) \(3+9+27+\cdots+3^{20}\) (b) \(\frac{2}{3}+\left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{3}+\cdots+\left(\frac{2}{3}\right)^{n}+\cdots\) (c) \((0.2)(10)+(0.2)(100)+(0.2)(1000)+\cdots\) (d) \(3+3(0.8)+3(0.8)^{2}+3(0.8)^{3}+\cdots\) (e) \((0.2)+(0.2)(1.3)+(0.2)(1.3)^{2}+(0.2)(1.3)^{3}+\cdots\) (f) \(1+x^{2}+x^{4}+x^{6}+\cdots\) for \(-1

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ \frac{1}{e}+\frac{2}{e^{2}}+\frac{4}{e^{3}}+\cdots+\frac{2^{n}}{e^{n+1}} $$

Make initial estimates to be sure that the answers you get are in the right ballpark. Suppose you are saving to buy some cattle. You plan to put \(\$ 200\) into an account every month for the next three years ( 36 deposits) to pay for the cows. You put your money into an account paying interest of \(4.5 \%\) per year compounded monthly. Immediately after the 36 th deposit, how much money will you have in your cattle fund?

Do the following. i. Write out the rst two terms of the series. ii. Determine whether or not the series converges. iii. If the series converges, determine its sum. $$ \sum_{n=3}^{\infty} \frac{(-1)^{n} 3}{2^{n}} $$

Give an example of each of the following. (a) An in nite series that converges and whose partial sums are always increasing (b) An in nite series that converges and whose partial sums oscillate around the sum of the series (c) An in nite series that diverges although its terms approach zero (d) An in nite series that diverges but whose partial sums do not grow without bound

Determine whether each of the following geometric series converges or diverges. If the series converges, determine to what it converges. (a) \(-\frac{4}{3}-\frac{1}{2}-\frac{3}{16}-\frac{9}{128}+\cdots\) (b) \(-\frac{1}{100}+\frac{1.1}{(100)^{2}}-\frac{1.21}{(100)^{3}}+\frac{1.331}{(100)^{4}}-\cdots\) (c) \(-\frac{7}{10000}+\frac{7}{11000}-\frac{7}{12100}+\frac{7}{13310}-\cdots\) (d) \(1-x+x^{2}-x^{3}+\cdots\) for \(|x|<1\)

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ -2+4-6+8-10+\cdots+16 $$

People who have slow metabolism due to a malfunctioning thyroid can take thyroid medication to alleviate their condition. For example, the boxer Muhammad Ali took Thyrolar 3, which is 3 grains of thyroid medication, every day. The amount of the drug in the bloodstream decays exponentially with time. The half-life of Thyrolar is 1 week. (a) Suppose one 3 -grain pill of Thyrolar is taken. Write an equation for the amount of the drug in the bloodstream \(t\) days after it has been taken. (Hint: In part (a) you are dealing with one 3 -grain pill of Thyrolar. Knowing the half-life of Thyrolar, you are asked to come up with a decay equation. This part of the problem has nothing to do with geometric sums.) (b) Suppose that Ali starts with none of the drug in his bloodstream. If he takes 3 grains of Thyrolar every day for ve days, how much Thyrolar is in his bloodstream immediately after having taken the fth pill? (c) Suppose Ali takes 3 grains of Thyrolar each day for one month. How much thyroid medication will be in his bloodstream right before he takes his 31 st pill? Right after? (d) After taking this medicine for many years, what was the amount of the drug in his body immediately after taking a pill? Historical note: Before one of his last ghts Muhammad Ali decided to up his dosage to 6 grains. In doing so he mimicked the symptoms of an overactive thyroid. The result in terms of the ght was dismal.

Do the following. i. Write out the rst two terms of the series. ii. Determine whether or not the series converges. iii. If the series converges, determine its sum. $$ \sum_{n=2}^{\infty} \frac{3^{n}}{4^{n-1}} $$

Write each of the following series rst as a repeating decimal and then as a fraction. (a) \(2+\frac{2}{10}+\frac{2}{100}+\frac{2}{1000}+\cdots\). (b) \(3+\frac{12}{10^{2}}+\frac{12}{10^{4}}+\frac{12}{10^{6}}+\cdots\)

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ -2+4-8+16-32+64 $$

Amanda, at the young age of 9, has gotten it rmly into her mind that she wants to be a doctor when she grows up. Her father, panic-stricken, wonders how the family will nance her college and medical school education. To assuage his anxiety he decides to set aside enough money in a bank account right now so that they will be able to withdraw \(\$ 10,000\) every year for eight years beginning nine years from today. Amanda s mother computes how much money they will have to put into the account with an annual interest rate of \(6 \%\) compounded quarterly. What gure should she arrive at?

Do the following. i. Write out the rst two terms of the series. ii. Determine whether or not the series converges. iii. If the series converges, determine its sum. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{3^{n}} $$

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ 0.2+0.06+0.018+0.0054+0.00162 $$

You have found a calling! You have some burning questions about elephants and want desperately to go to Kenya for a year. In addition to the plane fare you ll need some equipment, a guide, a jeep \(\ldots\) You ll need some money. You gure that you 11 need \(\$ 7000 .\) Each month beginning today you plan to put a xed amount of money into an account paying \(6 \%\) interest compounded monthly. How much must you deposit into the account each month if you plan to begin your eld work in four years?

Do the following. i. Write out the rst two terms of the series. ii. Determine whether or not the series converges. iii. If the series converges, determine its sum. $$ \sum_{n=100}^{\infty} \frac{10}{n} $$

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ 10+9.5+9+8.5+8+7.5+7+6.5 $$

A woman takes out a loan of \(\$ 100,000\) in order to nance a home. The interest rate is \(12 \%\) per year compounded monthly and she has a 30 -year mortgage. She will pay back the loan by paying a xed amount, \(M\) dollars, every month beginning one month from today and continuing for the next 30 years. (a) What is \(M ?\) (Hint: The sum of the present values of her 360 payments, pulled back to the present using an interest rate of \(12 \%\), should equal her loan.) (b) How much could she save each month if she could borrow at an interest rate of \(6.75 \%\) per year compounded monthly?

Do the following. i. Write out the rst two terms of the series. ii. Determine whether or not the series converges. iii. If the series converges, determine its sum. $$ \text { Does the series } \sum_{k=1}^{\infty} \frac{\ln (k+2)}{3 k} \text { converge or diverge? Explain. } $$

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ p+p^{3}+p^{5}+p^{7}+p^{9} $$

Mike L. and Mike C. have decided to establish the Mike and Mike Math Millenium Miracle Prize. The M\&M \(M^{3}\) prize is worth \(\$ 2000\) to the lucky winner. Due to limited funds, Mike and Mike have decided to award the prize once every 4 years, starting 10 years from now and going on inde nitely. (It s like the Fields Medal in Math, only more accessible.) They have begun to go door-to- door to take collections in order to establish the fund. How much money should the M\&M \(M^{3}\) Prize Fund contain right now in order to start payments 10 years from today? Assume a guaranteed interest rate of \(5 \%\) per year compounded annually.

Consider the sum $$ q^{5}-q^{7}+q^{9}-q^{11}+\cdots+q^{41} $$ (a) Put the sum into closed form. (b) Put the sum into summation notation. (c) Now put \(-q^{5}+q^{7}-q^{9}+q^{11}-\cdots-q^{41}\) into summation notation.

Express each of the sums in Problems 18 through 32 in closed form. Wherever possible, give a numerical approximation of the sum, rounded off to 3 decimal places. $$ 1-10+100-\cdots+10^{10} $$

Barry is thinking about buying a vehicle. He hears on the radio that he can buy a truck with no money down for two years and then make monthly payments of \(\$ 150 .\) He thinks this sounds good. He asks Angie if he should buy it. Angie says that she thinks he needs to know about the interest rates and how many years he ll have to make the monthly payments. Barry listens to the radio again and discovers that the monthly payments must be made for 10 years. He decides to compute the present value of the truck payments using an interest rate of \(6 \%\) per year compounded monthly or \(0.5 \%\) per month. What answer should he get?

For each of the following geometric sums, rst write the sum using summation notation and then write the sum in closed form. (a) \(\frac{2}{3^{2}}+\frac{2}{3^{4}}+\frac{2}{3^{6}}+\cdots \frac{2}{3^{18}}\) (b) \(1-2+2^{2}-2^{3}+2^{4}-\cdots+2^{46}\) (c) \(-\frac{1}{100}+\frac{1.1}{100}-\frac{1.21}{100}+\frac{1.331}{100}-\cdots-\frac{1.1^{100}}{100}\) (d) \(\frac{2}{3^{2}}+\frac{2^{2}}{3^{3}}+\frac{2^{3}}{34}+\cdots+\frac{2^{16}}{317}\)

Express each of the sums in closed form. Wherever possible, give a numerical approximation of the sum, rounded off to 3 decimal places. $$ \frac{3}{2}-\frac{3}{4}+\frac{3}{8}-\cdots-\frac{3}{2^{6}} $$

Brent and Rob were working on their math homework when Rob got a headache. Because Rob was incapacitated, Brent went to take a nap. Due to the headache he is blaming on the homework, Rob takes two aspirin. In the body aspirin metabolizes into salicylic acid, which has a half-life of two to three hours. (Source: The pharmacist at a CVS Pharmacy.) Rob is a big fellow, so for the purposes of this problem we ll say three hours is the half-life of salicylic acid. (a) The math headache is haunting him, so three hours later Rob takes two more aspirin. In fact, the headache is so bad that every three hours he takes two more aspirin. If he keeps this up inde nitely, will the level of salicylic acid in his body ever reach the level equivalent to taking four aspirin all at once? (b) Brent wakes up from a deep sleep, looks at his math homework, gets a headache, looks at Rob, and decides that he s going to take two aspirin every two hours. If he keeps this up inde nitely, will the level of salicylic acid in his body every reach the level equivalent to taking three aspirin all at once? Four aspirin all at once? Five aspirin all at once? (Assume again that the half-life of salicylic acid is three hours.)

Write the following without using summation notation and answer the following questions. (a) Is the series geometric? (b) Does the series converge? If so, indicate to what it converges. i) \(\sum_{n=0}^{\infty} \frac{3}{2}\left(\frac{5}{2}\right)^{n}\) ii) \(\sum_{n=0}^{\infty} \frac{15}{10^{2}}\left(\frac{1}{10}\right)^{3 n}\) iii) \(\sum_{n=1}^{\infty} \frac{3}{2}\left(\frac{-2}{3}\right)^{n}\) iv) \(\sum_{n=1}^{\infty} \ln n\)

Express each of the sums in closed form. Wherever possible, give a numerical approximation of the sum, rounded off to 3 decimal places. $$ \frac{2}{3}+2+6+\cdots+2(3)^{100} $$

Matt is saving money for his wedding. Suppose that at the beginning of every month he puts \(\$ 300\) in his savings account. The savings account gives interest of \(0.5 \%\) every month, for a nominal annual interest rate of \(6 \%\) per year compounded monthly. Matt does this for three years. How much will be in his savings account right after he makes the 36 th deposit?

Write the following sums in summation notation. In each case, determine the sum; i.e., sum the rst and determine what the second converges to. (a) \(500+500 e^{-1}+500 e^{2}+500 e^{3}+\cdots+500 e^{2}\) (b) \(\frac{5}{3}-\frac{5}{6}+\frac{5}{12}-\frac{5}{24}+\cdots\)

Express each of the sums in closed form. Wherever possible, give a numerical approximation of the sum, rounded off to 3 decimal places. $$ 2 e+2 e^{2}+2 e^{3}+\cdots+2 e^{12} $$

Nadia is saving for a trip to Venezuela. She estimates that she ll need \(\$ 3000\). She plans to put away a xed amount of money every month for the next 30 months so that immediately after the 30 th deposit she will have enough money for her trip. She puts her money into an account paying interest of \(4 \%\) per year compounded monthly. How much must she deposit every month? Before you begin calculations, do an estimate. Will she have to put aside more than \(\$ 100\) each month, or less?

Express each of the sums in closed form. Wherever possible, give a numerical approximation of the sum, rounded off to 3 decimal places. $$ 2 e+(2 e)^{2}+(2 e)^{3}+\cdots+(2 e)^{n} $$

Suppose you borrow \(\$ 18,000\) at an interest rate of \(8 \%\) compounded annually. You begin paying back money four years from today and make xed payments annually. You pay back the entire debt after six payments. What are your annual payments? Begin by guring out the ballpark gures. Will you pay more than \(\$ 3000\) each year? What is an upper bound for the amount of money you will pay each year?

Express each of the sums in closed form. Wherever possible, give a numerical approximation of the sum, rounded off to 3 decimal places. $$ -a+a^{2}-a^{3}+\cdots+(-a)^{n} $$

A prince takes out a loan of \(\$ 200,000\) in order to nance his castle. The interest rate is \(12 \%\) per year compounded monthly and he has a 15 -year mortgage. He will pay back the loan by paying a xed amount, \(F\) dollars, every month beginning one month from today and continuing for the next 15 years. What is \(F\) ? Note that the sum of the present values of his payments (pulled back to the present using an interest rate of \(12 \%\) ) should equal his loan.

Express each of the sums in closed form. Wherever possible, give a numerical approximation of the sum, rounded off to 3 decimal places. $$ \frac{1}{e}+\frac{2}{e^{2}}+\frac{4}{e^{3}}+\cdots+\frac{2^{n}}{e^{n+1}} $$

Express each of the sums in closed form. Wherever possible, give a numerical approximation of the sum, rounded off to 3 decimal places. $$ 5+15+45+\cdots+5 \cdot 3^{10} $$

Express each of the sums in closed form. Wherever possible, give a numerical approximation of the sum, rounded off to 3 decimal places. $$ \frac{2}{3}+\frac{1}{3}+\frac{1}{6}+\frac{1}{12}+\cdots+\frac{1}{3 \cdot 2^{100}} $$

At the beginning of each month a medical research center buries its refuse in its refuse dump. The monthly refuse deposit contains 40 grams of radioactive material. The radioactive material decays at a rate proportional to itself, with proportionality constant \(-0.2\) \(-\) (a) How much of the radioactive material buried at the beginning of the month is radioactive \(t\) months later? (b) Immediately after the 60 th monthly dump, how much radioactive material is in the refuse site? (c) If the situation goes on inde nitely, how much radioactive material will the site contain?

Express each of the sums in closed form. Wherever possible, give a numerical approximation of the sum, rounded off to 3 decimal places. $$ 1+\frac{9}{10}+\left(\frac{9}{10}\right)^{2}+\left(\frac{9}{10}\right)^{3}+\cdots+\left(\frac{9}{10}\right)^{n} $$

You take out a loan of \(\$ 3000\) at an interest rate of \(6 \%\) compounded monthly. You start paying back the loan exactly one year later. How much should each payment be if the loan is paid off after 24 equal monthly payments? Give an exact answer and an approximation correct to the nearest penny.

Express each of the sums in closed form. Wherever possible, give a numerical approximation of the sum, rounded off to 3 decimal places. $$ 1+\frac{11}{10}+\left(\frac{11}{10}\right)^{2}+\left(\frac{11}{10}\right)^{3}+\cdots+\left(\frac{11}{10}\right)^{200} $$

Express each of the sums in closed form. Wherever possible, give a numerical approximation of the sum, rounded off to 3 decimal places. $$ \frac{2}{p}+\frac{4}{p^{3}}+\frac{8}{p^{5}}+\frac{16}{p^{7}}+\cdots+\frac{2^{20}}{p^{39}} $$

Express each of the sums in closed form. Wherever possible, give a numerical approximation of the sum, rounded off to 3 decimal places. $$ m q+m^{2} q^{4}+m^{3} q^{7}+\cdots+m^{11} q^{31} $$

Express each of the sums in closed form. Wherever possible, give a numerical approximation of the sum, rounded off to 3 decimal places. \(\frac{1}{x}-\frac{1}{x^{2}}+\frac{1}{x^{3}}-\frac{1}{x^{4}}+\cdots-\frac{1}{x^{10}}\) Simplify your answer.

A ball is dropped from a height of 10 feet. Each time the ball bounces it rises to \(70 \%\) of its previous height. How far has the ball traveled when it hits the ground for the third time? For the 12 th time?