Chapter 18

For Problems 1 through 11, determine whether the series converges or diverges. If it converges, \(n d\) its sum. $$ 1-10+100-\cdots+(-10)^{n}+ $$

For Problems 1 through 17 determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ 1-10+100-\cdots+10^{10} $$

You have the choice of two awards. Award 1: You will receive six yearly payments of \(\$ 10,000\), the rst payment being made three years from today. Award 2: You will receive three payments of \(\$ 20,000\), the payments being made at two-year intervals, the rst payment being made two years from today. Suppose that the interest rate at the bank is \(4 \%\) per year compounded quarterly. (a) Find the present value of award 1 and the present value of award \(2 .\) Which present value is larger? Which award scheme would you choose? (b) Suppose you put each payment in the bank as soon as you receive it. How much money will be in the account eight years from today under the rst award scheme? Under the second award scheme?

For Problems 1 through 10, write the sum using summation notation. $$ 2^{3}+3^{3}+4^{3}+\cdots+19^{3} $$

For Problems 1 through 9, determine whether the series converges or diverges. Explain your reasoning. $$ 1-2+3-\cdots+(-1)^{(n+1) n}+\cdots $$

Determine whether the series converges or diverges. If it converges, \(n d\) its sum. $$ 0.3+0.03+0.003+0.0003+0.00003+\cdots $$

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ 0.3+0.33+0.333+0.3333+\cdots+0.333333333 $$

We ve been looking at banking because it ties in with the theoretical mathematics we are studying and because most of us have some sort of interface with a bank, be it only via an ATM card. The following questions are designed to direct your attention to the basics. (a) Suppose a bank compounds interest \(n\) times per year, \(n>1\). Which will be larger, the nominal interest rate, or the effective interest rate? (b) Suppose interest rates are xed at \(r \%\) per year compounded annually. Which is larger, the present value of \(\$ 1000\) in \(T\) years or \(\$ 1000\) in \((T+1)\) years? (c) Which is larger, the present value of \(\$ 1000\) in \(T\) years at a rate of \(4 \%\) compounded annually or the present value of \(\$ 1000\) in \(T\) years at a rate of \(5 \%\) compounded annually?

Write the sum using summation notation. $$ 2-3+4-5+6-\cdots+100 $$

For Problems , determine whether the series converges or diverges. Explain your reasoning. $$ \frac{1}{1000}+\frac{2}{1000}+\frac{3}{1000}+\frac{4}{1000}+\cdots $$

Determine whether the series converges or diverges. If it converges, \(n d\) its sum. $$ \frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\cdots+\frac{2}{3^{n}}+\cdots $$

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ 0.3+0.03+0.003+0.0003+\cdots+0.000000003 $$

(a) A friendly benefactor, impressed with Joselyn s enthusiasm for her mathematical studies, decides to award her a scholarship of \(\$ 6000\) to be paid to her ve years from today. How much money must the benefactor put aside today in an account earning a nominal annual interest of \(4 \%\) compounded continuously in order to cover Joselyn s award? (This question asks what is the present value of \(\$ 6000\) in ve years at an interest rate of \(4 \%\) compounded continuously? ) (b) Another benefactor, interested in Patrick s potential, promises Pat that if he continues his studies in mathematics he will be awarded a scholarship. The scholarship will given in three payments of \(\$ 2000\), the rst payment being made in three years (when he graduates), the second in four years, and the last payment being made ve years from today. The benefactor has put aside money for Pat s scholarship in an account earning \(4 \%\) nominal annual interest compounded continuously in order to cover Pat s award. Pat says that he will promise to take mathematics, but he would like his scholarship money up front and immediately. Because the benefactor has not set aside the full \(\$ 6000\) now, he agrees to give Pat the present value of the award, i.e., the amount of money he has set aside in an account for Pat. How much money should Pat be expecting? (Hint: You need to do three separate calculations. Find the present value of the rst payment, then of the second, and then of the third.) (c) Which answer did you expect to be bigger, the answer to part (a) or the answer to part (b)? Why? Have your calculations matched your expectations?

Write the sum using summation notation. $$ 2^{3}+3^{4}+4^{5}+\cdots+100^{101} $$

For Problems , determine whether the series converges or diverges. Explain your reasoning. $$ \frac{2}{3}+\frac{2}{4}+\frac{2}{5}+\cdots+\frac{2}{n}+\cdots $$

Determine whether the series converges or diverges. If it converges, \(n d\) its sum. $$ \frac{2}{3}+2+6+\cdots+2(3)^{n}+\cdots $$

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ \frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\cdots+\frac{2}{6561} $$

A ball is thrown from the ground to a height of 16 feet. Each time the ball bounces it rises up to \(60 \%\) of its previous height. What is the total distance traveled by the ball? (Hint: Keep in mind that between every bounce the ball is going up and then coming back down.)

Write the sum using summation notation. (a) \(4 x^{2}+4 x^{3}+4 x^{4}+4 x^{5}+\cdots+\) (b) \(2 x+3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+\cdots+\)

For Problems , determine whether the series converges or diverges. Explain your reasoning. \(\frac{1}{2 \cdot 2^{2}}+\frac{1}{3 \cdot 2^{3}}+\frac{1}{4 \cdot 2^{4}}+\frac{1}{5 \cdot 2^{5}}+\cdots\) (Hint: Compare this term-by-term to a geometric series you know. Choose a convergent geometric series whose terms are larger than the terms of this series.)

Determine whether the series converges or diverges. If it converges, \(n d\) its sum. $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}+\cdots $$

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ \frac{2}{3}+2+6+\cdots+2(3)^{100} $$

Make initial estimates to be sure that the answers you get are in the right ballpark. Suppose you borrow \(\$ 10,000\) at an interest rate of \(7 \%\) compounded annually. You begin paying back money one year from today and make uniform payments of \(\$ P\) annually. You pay back the entire debt after 10 payments. What are your annual payments? Hint: Pull each of the 10 payments of \(\$ P\) back to the present. The sum should be \(\$ 10,000\). Ballpark gures: If no interest were charged, then you would pay \(\$ 10,000 / 10=\) \(\$ 1000\). If you had to pay interest on the entire \(\$ 10,000\) for 10 years, then you d pay \(\$ 10,000 \cdot 1.07^{10} / 10 .\) The actual answer is somewhere between these two extremes.

Write the sum using summation notation. $$ 1-10+100-\cdots+(-10)^{n}+\cdots $$

For Problems , determine whether the series converges or diverges. Explain your reasoning. \(\frac{(\sin 1)^{2}}{3}+\frac{(\sin 2)^{2}}{3^{2}}+\frac{(\sin 3)^{2}}{3^{3}}+\cdots+\frac{(\sin n)^{2}}{3^{n}}+\cdots\) (Hint: Compare this term-by-term to a geometric series you know. Choose a convergent geometric series whose terms are larger than the terms of this series.)

Determine whether the series converges or diverges. If it converges, \(n d\) its sum. $$ \frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\frac{1}{32}+\cdots $$

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{100} $$

Make initial estimates to be sure that the answers you get are in the right ballpark. Suppose you borrow \(\$ 10,000\) at an interest rate of \(6 \%\) compounded annually. You begin paying back the loan one year from today and make uniform payments annually. You pay back the entire debt after 8 payments. What are your annual payments?

Write the sum using summation notation. $$ 0.3+0.03+0.003+0.0003+0.00003+\cdots $$

For Problems , determine whether the series converges or diverges. Explain your reasoning. $$ \frac{3^{3}}{3}+\frac{4^{4}}{4}+\frac{5^{5}}{5}+\cdots $$

Determine whether the series converges or diverges. If it converges, \(n d\) its sum. $$ \frac{2}{3}+1+\frac{3}{2}+\frac{9}{4}+\cdots $$

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots+\frac{1}{64} $$

Make initial estimates to be sure that the answers you get are in the right ballpark. Suppose you borrow \(\$ 10,000\) at an interest rate of \(7 \%\) compounded annually. You begin paying back money ve years from today and make payments annually. You pay back the entire debt after 10 payments. What are your xed annual payments?

Write the sum using summation notation. (a) \(\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\cdots+\frac{2}{3^{n}}+\cdots\) (b) \(\frac{2}{3}+2+6+\cdots+2(3)^{n}+\cdots\)

For Problems , determine whether the series converges or diverges. Explain your reasoning. $$ \frac{-1}{2}-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\cdots-\frac{1}{n}-\cdots $$

Determine whether the series converges or diverges. If it converges, \(n d\) its sum. $$ \frac{3}{2}-\frac{3}{4}+\frac{3}{8}-\cdots+\frac{(-1)^{n+1} 3}{2^{n}}+\cdots $$

Make initial estimates to be sure that the answers you get are in the right ballpark. Suppose you borrow some money at an interest rate of \(6 \%\) compounded annually. You begin paying back one year from today and make payments annually. You pay back the entire debt after 10 payments of \(\$ 1000\) each. How much money did you borrow?

Write the sum using summation notation. (a) \(1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}+\cdots\) (b) \(\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\frac{1}{32}+\cdots\)

For Problems , determine whether the series converges or diverges. Explain your reasoning. $$ \frac{1}{2}-\frac{1}{2}+\frac{1}{2}-\frac{1}{2}+\cdots $$

Determine whether the series converges or diverges. If it converges, \(n d\) its sum. $$ e+1+e^{-1}+e^{-2}+e^{-3}+\cdots $$

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ \frac{3}{2}-\frac{3}{4}+\frac{3}{8}-\cdots-\frac{3}{2^{6}} $$

Make initial estimates to be sure that the answers you get are in the right ballpark. Suppose you borrow some money at an interest rate of \(6 \%\) compounded monthly. You begin paying back one month from today and make payments monthly. You pay back the entire debt after 180 payments of \(\$ 1000\) each. (This is a 15 -year mortgage.) How much money did you borrow?

For Problems , determine whether the series converges or diverges. Explain your reasoning. $$ (1+1)^{1}+\left(1+\frac{1}{2}\right)^{2}+\left(1+\frac{1}{3}\right)^{3}+\cdots+\left(1+\frac{1}{n}\right)^{n}+\cdots $$

Determine whether the series converges or diverges. If it converges, \(n d\) its sum. $$ 2 e+2 e^{2}+2 e^{3}+\cdots+2 e^{n}+\cdots $$

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ \frac{3}{2}-\frac{4}{4}+\frac{5}{8}-\cdots-\frac{8}{2^{6}} $$

Make initial estimates to be sure that the answers you get are in the right ballpark. Suppose you borrow some money at an interest rate of \(6 \%\) compounded monthly. You begin paying back one year from today and make payments annually. You pay back the entire debt after 30 payments of \(\$ 1000\) each. How much money did you borrow?

Write the sum using summation notation. (a) \(e+1+e^{-1}+e^{-2}+e^{-3}+\cdots\) (b) \(2 e+2 e^{2}+2 e^{3}+\cdots+2 e^{n}+\cdots\) (c) \((2 e)^{-2}+(2 e)^{-3}+(2 e)^{-4}+\cdots+(2 e)^{-n}+\cdots\)

Determine whether the series converges or diverges. If it converges, \(n d\) its sum. $$ (2 e)^{-2}+(2 e)^{-3}+(2 e)^{-4}+\cdots+(2 e)^{-n}+\cdots $$

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ -a+a^{2}-a^{3}+\cdots+(-a)^{17} $$

Make initial estimates to be sure that the answers you get are in the right ballpark. Suppose you are saving for a big trip abroad. You estimate that you 11 need \(\$ 4000 .\) You plan to put away a xed amount of money every month for the next two years \((24\) deposits) so that immediately after the 24 th deposit you have enough money for your trip. You put your money into an account paying interest of \(4.5 \%\) per year compounded monthly. How much must you deposit every month?