Answer: ?When money is spent on goods and services, those who receive the money also

Explain what it means to say that .

3–4 Calculate the sum of the series whose partial sums are given.

3-4 Calculate the sum of the series \(\sum_{n=1}^{\infty} a_{n}\) whose partial sums are given. \(s_{n}=\frac{n^{2}-1}{4 n^{2}+1}\) Text Transcription: sum_n=1 ^infty a_n s_n = n^2 - 1 / 4n^2 + 1

5–8 Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?

5–8 Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?

5–8 Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?

5–8 Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?

9–14 Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.

9–14 Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.

9–14 Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.

9–14 Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.

9–14 Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why. \(\sum_{n=2}^{\infty} \frac{1}{n(n+2)}\) Text Transcription: \sum_{n=2}^{\infty} \frac{1}{n(n+2)}

Let . (a) Determine whether is convergent. (b) Determine whether is convergent.

(a) Explain the difference between (b) Explain the difference between

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 25.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 26.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum.

Explain what it means to say that .

3–4 Calculate the sum of the series whose partial sums are given.

3-4 Calculate the sum of the series \(\sum_{n=1}^{\infty} a_{n}\) whose partial sums are given. \(s_{n}=\frac{n^{2}-1}{4 n^{2}+1}\) Text Transcription: sum_n=1 ^infty a_n s_n = n^2 - 1 / 4n^2 + 1

5–8 Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?

5–8 Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?

5–8 Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?

5–8 Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?

9–14 Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.

9–14 Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.

9–14 Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.

9–14 Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.

9–14 Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why. \(\sum_{n=2}^{\infty} \frac{1}{n(n+2)}\) Text Transcription: \sum_{n=2}^{\infty} \frac{1}{n(n+2)}

Let . (a) Determine whether is convergent. (b) Determine whether is convergent.

(a) Explain the difference between (b) Explain the difference between

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 25.

17–26 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 26.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum. 35.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum. 36.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum. 37.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum. 38.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum. 39.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum. 40.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum. 41.

27–42 Determine whether the series is convergent or divergent. If it is convergent, find its sum. 42.

43–48 Determine whether the series is convergent or divergent by expressing as a telescoping sum (as in Example 7). If it is convergent, find its sum. 43.

43–48 Determine whether the series is convergent or divergent by expressing as a telescoping sum (as in Example 7). If it is convergent, find its sum. 44.

43–48 Determine whether the series is convergent or divergent by expressing as a telescoping sum (as in Example 7). If it is convergent, find its sum. 45.

43–48 Determine whether the series is convergent or divergent by expressing as a telescoping sum (as in Example 7). If it is convergent, find its sum. 46.

43–48 Determine whether the series is convergent or divergent by expressing as a telescoping sum (as in Example 7). If it is convergent, find its sum. 47.

43–48 Determine whether the series is convergent or divergent by expressing as a telescoping sum (as in Example 7). If it is convergent, find its sum. 48.

Let …. (a) Do you think that or ? (b) Sum a geometric series to find the value of . (c) How many decimal representations does the number 1 have? (d) Which numbers have more than one decimal representation?

A sequence of terms is defined by Calculate .

51–56 Express the number as a ratio of integers. 51.

51–56 Express the number as a ratio of integers. 52.

51–56 Express the number as a ratio of integers. 53.

51–56 Express the number as a ratio of integers. 54.

51–56 Express the number as a ratio of integers. 55.

51–56 Express the number as a ratio of integers. 56.

57–63 Find the values of for which the series converges. Find the sum of the series for those values of . 57.

57–63 Find the values of for which the series converges. Find the sum of the series for those values of . 58.

57–63 Find the values of for which the series converges. Find the sum of the series for those values of . 59.

57–63 Find the values of for which the series converges. Find the sum of the series for those values of . 60.

57–63 Find the values of for which the series converges. Find the sum of the series for those values of . 61.

57–63 Find the values of for which the series converges. Find the sum of the series for those values of . 62.

57–63 Find the values of for which the series converges. Find the sum of the series for those values of . 63.

We have seen that the harmonic series is a divergent series whose terms approach 0. Show that is another series with this property.

65–66 Use the partial fraction command on your CAS to find a convenient expression for the partial sum, and then use this expression to find the sum of the series. Check your answer by using the CAS to sum the series directly. 65.

65–66 Use the partial fraction command on your CAS to find a convenient expression for the partial sum, and then use this expression to find the sum of the series. Check your answer by using the CAS to sum the series directly. 66.

If the th partial sum of a series is find and .

If the th partial sum of a series is , find and .

After injection of a dose of insulin, the concentration of insulin in a patient’s system decays exponentially and so it can be written as , where represents time in hours and is a positive constant. (a) If a dose is injected every hours, write an expression for the sum of the residual concentrations just before the st injection. (b) Determine the limiting pre-injection concentration. (c) If the concentration of insulin must always remain at or above a critical value , determine a minimal dosage in terms of , , and .

When money is spent on goods and services, those who receive the money also spend some of it. The people receiving some of the twice-spent money will spend some of that, and so on. Economists call this chain reaction the multiplier effect. In a hypothetical isolated community, the local government begins the process by spending dollars. Suppose that each recipient of spent money spends 100c% and saves 100c% of the money that he or she receives. The values c and s are called the marginal propensity to consume and the marginal propensity to save and, of course, . (a) Let be the total spending that has been generated after transactions. Find an equation for . (b) Show that , where . The number is called the multiplier. What is the multiplier if the marginal propensity to consume is 80%? Note: The federal government uses this principle to justify deficit spending. Banks use this principle to justify lending a large percentage of the money that they receive in deposits.

A certain ball has the property that each time it falls from a height onto a hard, level surface, it rebounds to a height , where . Suppose that the ball is dropped from an initial height of meters. (a) Assuming that the ball continues to bounce indefinitely, find the total distance that it travels. (b) Calculate the total time that the ball travels. (Use the fact that the ball falls meters in seconds.) (c) Suppose that each time the ball strikes the surface with velocity it rebounds with velocity , where . How long will it take for the ball to come to rest?

Find the value of if

Find the value of such that

In Example 8 we showed that the harmonic series is divergent. Here we outline another method, making use of the fact that for any . (See Exercise 4.3.78.) If is the th partial sum of the harmonic series, show that . Why does this imply that the harmonic series is divergent?

Graph the curves , , for on a common screen. By finding the areas between successive curves, give a geometric demonstration of the fact, shown in Example 7, that

The figure shows two circles and of radius 1 that touch at . is a common tangent line; is the circle that touches , , and ; is the circle that touches , , and ; is the circle that touches , , and . This procedure can be continued indefinitely and produces an infinite sequence of circles . Find an expression for the diameter of and thus provide another geometric demonstration of Example 7.

A right triangle is given with and . is drawn perpendicular to , is drawn perpendicular to , , and this process is continued indefinitely, as shown in the figure. Find the total length of all the perpendiculars in terms of and .

What is wrong with the following calculation? (Guido Ubaldus thought that this proved the existence of God because “something has been created out of nothing.”)

Suppose that is known to be a convergent series. Prove that is a divergent series.

If is divergent and , show that is divergent.

If is convergent and is divergent, show that the series is divergent. [Hint: Argue by contradiction.]

If and are both divergent, is necessarily divergent?

Suppose that a series has positive terms and its partial sums satisfy the inequality for all . Explain why must be convergent.

The Fibonacci sequence was defined in Section 11.1 by the equations Show that each of the following statements is true. (a) (b) (c)

The Cantor set, named after the German mathematician Georg Cantor (1845–1918), is constructed as follows. We start with the closed interval [0, 1] and remove the open interval . That leaves the two intervals and and we remove the open middle third of each. Four intervals remain and again we remove the open middle third of each of them. We continue this procedure indefinitely, at each step removing the open middle third of every interval that remains from the preceding step. The Cantor set consists of the numbers that remain in [0, 1] after all those intervals have been removed. (a) Show that the total length of all the intervals that are removed is 1. Despite that, the Cantor set contains infinitely many numbers. Give examples of some numbers in the Cantor set. (b) The Sierpinski carpet is a two-dimensional counterpart of the Cantor set. It is constructed by removing the center one-ninth of a square of side 1, then removing the centers of the eight smaller remaining squares, and so on. (The figure shows the first three steps of the construction.) Show that the sum of the areas of the removed squares is 1. This implies that the Sierpinski carpet has area 0.

(a) A sequence is defined recursively by the equation for , where and can be any real numbers. Experiment with various values of and and use your calculator to guess the limit of the sequence. (b) Find in terms of and by expressing in terms of and summing a series.

Consider the series . (a) Find the partial sums and . Do you recognize the denominators? Use the pattern to guess a formula for . (b) Use mathematical induction to prove your guess. (c) Show that the given infinite series is convergent, and find its sum.

In the figure there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length 1, find the total area occupied by the circles.

(a) What is the difference between a sequence and a series? (b) What is a convergent series? What is a divergent series?

A patient takes 150 mg of a drug at the same time every day. Just before each tablet is taken, 5% of the drug remains in the body. (a) What quantity of the drug is in the body after the third tablet? After the nth tablet? (b) What quantity of the drug remains in the body in the long run?

Prove part (i) of Theorem 8.