Answer: In Exercises 7986, find all values of x for which the series converges. For

In Exercises 16, find the first five terms of the sequence of partial sums.1 1 4 1 9 1 16 1 25 . . . Exer

In Exercises 16, find the first five terms of the sequence of partial sums.

In Exercises 16, find the first five terms of the sequence of partial sums.3  9 2 27 4  81 8 243 16  . .

In Exercises 16, find the first five terms of the sequence of partial sums.

In Exercises 16, find the first five terms of the sequence of partial sums.

In Exercises 16, find the first five terms of the sequence of partial sums.
n
1 1n1 n

In Exercises 716, verify that the infinite series diverges
n
0 3 3 2 n

In Exercises 716, verify that the infinite series diverges
n
0  4 3 n

In Exercises 716, verify that the infinite series diverges

In Exercises 716, verify that the infinite series diverges

In Exercises 716, verify that the infinite series diverges

In Exercises 716, verify that the infinite series diverges

In Exercises 716, verify that the infinite series diverges
2 1 n
1 n2 n2 1

In Exercises 716, verify that the infinite series diverges
n
1 n n
2 1

In Exercises 716, verify that the infinite series diverges
n n
1 2n 1 2n1

In Exercises 716, verify that the infinite series diverges
n
1 n! 2

In Exercises 1722, match the series with the graph of its sequence of partial sums. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] Use the graph to estimate the sum of the series. Confirm your answer analytically

In Exercises 1722, match the series with the graph of its sequence of partial sums. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] Use the graph to estimate the sum of the series. Confirm your answer analytically

In Exercises 1722, match the series with the graph of its sequence of partial sums. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] Use the graph to estimate the sum of the series. Confirm your answer analytically
n
0 15 4  1 4

In Exercises 1722, match the series with the graph of its sequence of partial sums. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] Use the graph to estimate the sum of the series. Confirm your answer analytically
n
0 17 3  8 9 n

In Exercises 1722, match the series with the graph of its sequence of partial sums. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] Use the graph to estimate the sum of the series. Confirm your answer analytically
n
0 17 3 1 2

In Exercises 1722, match the series with the graph of its sequence of partial sums. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] Use the graph to estimate the sum of the series. Confirm your answer analytically
n
0  2 5 n

In Exercises 2328, verify that the infinite series converges.
n
1 1 nn 1 Use partial fractions.

In Exercises 2328, verify that the infinite series converges.

In Exercises 2328, verify that the infinite series converges.

In Exercises 2328, verify that the infinite series converges.

In Exercises 2328, verify that the infinite series converges.
n
0 0.9 n
1 0.9 0.81 0.729 . . .

In Exercises 2328, verify that the infinite series converges.
n
0 0.9 n
1 0.9 0.81 0.729 . . .

Numerical, Graphical, and Analytic Analysis In Exercises 2934, (a) find the sum of the series, (b) use a graphing utility to find the indicated partial sum and complete the table, (c) use a graphing utility to graph the first 10 terms of the sequence of partial sums and a horizontal line representing the sum, and (d) explain the relationship between the magnitudes of the terms of the series and the rate at which the sequence of partial sums approaches the sum of the series.
nn 4 n
1 6 nn 3 Sn

Numerical, Graphical, and Analytic Analysis In Exercises 2934, (a) find the sum of the series, (b) use a graphing utility to find the indicated partial sum and complete the table, (c) use a graphing utility to graph the first 10 terms of the sequence of partial sums and a horizontal line representing the sum, and (d) explain the relationship between the magnitudes of the terms of the series and the rate at which the sequence of partial sums approaches the sum of the series.

Numerical, Graphical, and Analytic Analysis In Exercises 2934, (a) find the sum of the series, (b) use a graphing utility to find the indicated partial sum and complete the table, (c) use a graphing utility to graph the first 10 terms of the sequence of partial sums and a horizontal line representing the sum, and (d) explain the relationship between the magnitudes of the terms of the series and the rate at which the sequence of partial sums approaches the sum of the series.

Numerical, Graphical, and Analytic Analysis In Exercises 2934, (a) find the sum of the series, (b) use a graphing utility to find the indicated partial sum and complete the table, (c) use a graphing utility to graph the first 10 terms of the sequence of partial sums and a horizontal line representing the sum, and (d) explain the relationship between the magnitudes of the terms of the series and the rate at which the sequence of partial sums approaches the sum of the series.
n
1 30.85n1

Numerical, Graphical, and Analytic Analysis In Exercises 2934, (a) find the sum of the series, (b) use a graphing utility to find the indicated partial sum and complete the table, (c) use a graphing utility to graph the first 10 terms of the sequence of partial sums and a horizontal line representing the sum, and (d) explain the relationship between the magnitudes of the terms of the series and the rate at which the sequence of partial sums approaches the sum of the series.
n
1 100.25 n1

Numerical, Graphical, and Analytic Analysis In Exercises 2934, (a) find the sum of the series, (b) use a graphing utility to find the indicated partial sum and complete the table, (c) use a graphing utility to graph the first 10 terms of the sequence of partial sums and a horizontal line representing the sum, and (d) explain the relationship between the magnitudes of the terms of the series and the rate at which the sequence of partial sums approaches the sum of the series.
n
1 5 1 3 n1

In Exercises 3550, find the sum of the convergent series
nn 2 n
2 1 n2  1

In Exercises 3550, find the sum of the convergent series n
1 4
nn 2

In Exercises 3550, find the sum of the convergent series

In Exercises 3550, find the sum of the convergent series

In Exercises 3550, find the sum of the convergent series
n
0  1 2

In Exercises 3550, find the sum of the convergent series

In Exercises 3550, find the sum of the convergent series

In Exercises 3550, find the sum of the convergent series n
0 2  2 3

In Exercises 3550, find the sum of the convergent series 0.1 0.01 0.001 . . .

In Exercises 3550, find the sum of the convergent series 8 6 9 2 27 8 . . . 1

In Exercises 3550, find the sum of the convergent series

In Exercises 3550, find the sum of the convergent series 4  2 1  1 2 . . . 3

In Exercises 3550, find the sum of the convergent series
 n
0  1 2n  1

In Exercises 3550, find the sum of the convergent series
n
1 0.7 n 0.9 n
 n

In Exercises 3550, find the sum of the convergent series
2 3n  2 n
1 sin 1n

In Exercises 3550, find the sum of the convergent series

In Exercises 5156, (a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers. 0.4

In Exercises 5156, (a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers. 0.9

In Exercises 5156, (a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers. 0.81

In Exercises 5156, (a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers. 0.01

In Exercises 5156, (a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers. 0.075

In Exercises 5156, (a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers.0.215

In Exercises 5772, determine the convergence or divergence of the series
2n  1 n
1 n 10 10n 1 0.

In Exercises 5772, determine the convergence or divergence of the series

In Exercises 5772, determine the convergence or divergence of the series

In Exercises 5772, determine the convergence or divergence of the series n
1 1
nn 3

In Exercises 5772, determine the convergence or divergence of the series

In Exercises 5772, determine the convergence or divergence of the series

In Exercises 5772, determine the convergence or divergence of the series
n n
0 4 2n

In Exercises 5772, determine the convergence or divergence of the series

In Exercises 5772, determine the convergence or divergence of the series

In Exercises 5772, determine the convergence or divergence of the series
n
1 2n
100

In Exercises 5772, determine the convergence or divergence of the series
n n
2 n ln n

In Exercises 5772, determine the convergence or divergence of the series

In Exercises 5772, determine the convergence or divergence of the series

In Exercises 5772, determine the convergence or divergence of the series
n
1 en

In Exercises 5772, determine the convergence or divergence of the series

In Exercises 5772, determine the convergence or divergence of the series
n
1 ln n 1 n
 n

State the definitions of convergent and divergent series.

Describe the difference between and
n
1 an
5.

Define a geometric series, state when it converges, and give the formula for the sum of a convergent geometric series.

State the Term Test for Divergence.

Let Discuss the convergence of and an

Explain any differences among the following series. (a) (b) (c)
n
1
ak

In Exercises 7986, find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x .
n
1 xn 2n

In Exercises 7986, find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x .

In Exercises 7986, find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x .

In Exercises 7986, find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x .
n
0 4 x  3 4  n

In Exercises 7986, find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x .
n
0 1n xn

In Exercises 7986, find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x .

In Exercises 7986, find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x .
n
0  1 x n

In Exercises 7986, find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x .
n
1  x2 x2 4 n

(a) You delete a finite number of terms from a divergent series. Will the new series still diverge? Explain your reasoning. (b) You add a finite number of terms to a convergent series. Will the new series still converge? Explain your reasoning.

Think About It Consider the formula Given and can you conclude that either of the following statements is true? Explain your reasoning.

In Exercises 89 and 90, (a) find the common ratio of the geometric series, (b) write the function that gives the sum of the series, and (c) use a graphing utility to graph the function and the partial sums and What do you notice?

In Exercises 89 and 90, (a) find the common ratio of the geometric series, (b) write the function that gives the sum of the series, and (c) use a graphing utility to graph the function and the partial sums and What do you notice?

In Exercises 91 and 92, use a graphing utility to graph the function. Identify the horizontal asymptote of the graph and determine its relationship to the sum of the series.

In Exercises 91 and 92, use a graphing utility to graph the function. Identify the horizontal asymptote of the graph and determine its relationship to the sum of the series.

Writing In Exercises 93 and 94, use a graphing utility to determine the first term that is less than 0.0001 in each of the convergent series. Note that the answers are very different. Explain how this will affect the rate at which the series converges.

Writing In Exercises 93 and 94, use a graphing utility to determine the first term that is less than 0.0001 in each of the convergent series. Note that the answers are very different. Explain how this will affect the rate at which the series converges.

Marketing An electronic games manufacturer producing a new product estimates the annual sales to be 8000 units. Each year 10% of the units that have been sold will become inoperative. So, 8000 units will be in use after 1 year, units will be in use after 2 years, and so on. How many units will be in use after years?

Depreciation A company buys a machine for $225,000 that depreciates at a rate of 30% per year. Find a formula for the value of the machine after years. What is its value after 5 years?

Multiplier Effect The annual spending by tourists in a resort city is $100 million. Approximately 75% of that revenue is again spent in the resort city, and of that amount approximately 75% is again spent in the same city, and so on. Write the geometric series that gives the total amount of spending generated by the $100 million and find the sum of the series.

Multiplier Effect Repeat Exercise 97 if the percent of the revenue that is spent again in the city decreases to 60%.

Distance A ball is dropped from a height of 16 feet. Each time it drops feet, it rebounds 0.81 feet. Find the total distance traveled by the ball.

Time The ball in Exercise 99 takes the following times for each fall. Beginning with the ball takes the same amount of time to bounce up as it does to fall, and so the total time elapsed before it comes to rest is given by Find this total time.

Probability In Exercises 101 and 102, the random variable represents the number of units of a product sold per day in a store. The probability distribution of is given by Find the probability that two units are sold in a given day and show that P1 P2 P3 . . . 1. [P

Probability In Exercises 101 and 102, the random variable represents the number of units of a product sold per day in a store. The probability distribution of is given by Find the probability that two units are sold in a given day and show that P1 P2 P3 . . . 1. [P

Probability A fair coin is tossed repeatedly. The probability that the first head occurs on the toss is given by where (a) Show that (b) The expected number of tosses required until the first head occurs in the experiment is given by Is this series geometric? (c) Use a computer algebra system to find the sum in part (b).

Probability In an experiment, three people toss a fair coin one at a time until one of them tosses a head. Determine, for each person, the probability that he or she tosses the first head. Verify that the sum of the three probabilities is 1.

Area The sides of a square are 16 inches in length. A new square is formed by connecting the midpoints of the sides of the original square, and two of the triangles outside the second square are shaded (see figure). Determine the area of the shaded regions (a) if this process is continued five more times and (b) if this pattern of shading is continued infinitely

Length A right triangle is shown above where and Line segments are continually drawn to be perpendicular to the triangle, as shown in the figure. (a) Find the total length of the perpendicular line segments in terms of z and (b) If and find the total length of the perpendicular line segments.

In Exercises 107110, use the formula for the partial sum of a geometric series
n
1 i0 ari a1
rn

In Exercises 107110, use the formula for the partial sum of a geometric series
n
1 i0 ari a1
rn

In Exercises 107110, use the formula for the partial sum of a geometric series
n
1 i0 ari a1
rn Salary You go to work at a company that pays $0.01 for the first day, $0.02 for the second day, $0.04 for the third day, and so on. If the daily wage keeps doubling, what would your total income be for working (a) 29 days, (b) 30 days, and (c) 31 days?

In Exercises 107110, use the formula for the partial sum of a geometric series
n
1 i0 ari a1
rn

Annuities In Exercises 111114, consider making monthly deposits of dollars in a savings account at an annual interest rate Use the results of Exercise 110 to find the balance after years if the interest is compounded (a) monthly and (b) continuously.

Annuities In Exercises 111114, consider making monthly deposits of dollars in a savings account at an annual interest rate Use the results of Exercise 110 to find the balance after years if the interest is compounded (a) monthly and (b) continuously.

Annuities In Exercises 111114, consider making monthly deposits of dollars in a savings account at an annual interest rate Use the results of Exercise 110 to find the balance after years if the interest is compounded (a) monthly and (b) continuously.

Annuities In Exercises 111114, consider making monthly deposits of dollars in a savings account at an annual interest rate Use the results of Exercise 110 to find the balance after years if the interest is compounded (a) monthly and (b) continuously.

Modeling Data The annual sales (in millions of dollars) for Avon Products, Inc. from 1993 through 2002 are given below as ordered pairs of the form where represents the year, with corresponding to 1993. (Source: 2002 Avon Products, Inc. Annual Report)) Use the regression capabilities of a graphing utility to find a model of the form 4, 5, 12 for the data. Graphically compare the points and the model. (b) Use the data to find the total sales for the 10-year period. (c) Approximate the total sales for the 10-year period using the formula for the sum of a geometric series. Compare the result with that in part (b).

Salary You accept a job that pays a salary of $40,000 for the first year. During the next 39 years you receive a 4% raise each year. What would be your total compensation over the 40-year period?

True or False? In Exercises 117122, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. f then converges.

True or False? In Exercises 117122, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

True or False? In Exercises 117122, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

True or False? In Exercises 117122, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The series diverges.

True or False? In Exercises 117122, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

True or False? In Exercises 117122, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Every decimal with a repeating pattern of digits is a rational number.

Show that the series can be written in the telescoping form where and is the partial sum

Let be a convergent series, and let be the remainder of the series after the first terms. Prove that im N RN
0.

Find two divergent series and such that converges.

Given two infinite series and such that converges and diverges, prove that diverges.

Suppose that diverges and c is a nonzero constant. Prove that diverges.

If converges where is nonzero, show that diverges

The Fibonacci sequence is defined recursively by where and (a) Show that (b) Show that
n
0 1 an1 an3
1. 1 an1

Find the values of for which the infinite series converges. What is the sum when the series converges?

Prove that if converges to and then there exists a number such that for n > n r 1 r2 1 r3 . . .
1 r  1 ,

Writing The figure below represents an informal way of showing that Explain how the figure implies this conclusion. FOR FURTHER INFORMATION For more on this exercise, see the article Convergence with Pictures by P.J. Rippon in American Mathematical Monthly.

Writing Read the article The Exponential-Decay Law Applied to Medical Dosages by Gerald M. Armstrong and Calvin P. Midgley in Mathematics Teacher. (To view this article, go to the website www.matharticles.com.) Then write a paragraph on how a geometric sequence can be used to find the total amount of a drug that remains in a patients system after equal doses have been administered (at equal time intervals). n

Write as a rational number.

1, 2, 2, 3, 3, 4, . . . , where the nth term is given by Show that if x and y are positive integers and then The following procedure shows how to make a table disappear by removing only half of the table! (a) Original table has a length of (b) Remove of the table centered at the midpoint. Each remaining piece has a length that is less than (c) Remove of the table by taking sections of length from the centers of each of the two remaining pieces. Now, you have removed of the table. Each remaining piece has a length that is less than (d) Remove of the table by taking sections of length from the centers of each of the four remaining pieces. Now, you have removed of the table. Each remaining piece has a length that is less than Will continuing this process cause the table to disappear, even though you have only removed half of the table? Why? FOR FURTHER INFORMATION Read the article Cantors Disappearing Table by Larry E. Knop in The College Mathematics Journal. To view this article, go to the website www.matharticles.com. 1 8L. Mathematical Association of America. All rights reserved.