Telescoping series For the following

Geometric sums Evaluate the following geometric sums. \(\sum_{k=0}^{9}\left(-\frac{3}{4}\right)^{k}\)

7-18. Geometric sums Evaluate the following geometric sums. \(\frac{1}{4}+\frac{1}{12}+\frac{1}{36}+\frac{1}{108}+\cdots+\frac{1}{2916}\)

Geometric sums Evaluate the following geometric sums. \(\sum _{k=0}^{20}\ (-1)^k\)

Snowflake island fractal The fractal called the snowflake island (or Koch island) is constructed as follows: Let \(I_{0}\) be an equilateral triangle with sides of length 1. The figure \(I_{1}\) is obtained by replacing the middle third of each side of \(I_{0}\) by a new outward equilateral triangle with sides of length 1/3 (see figure). The process is repeated where \(I_{n+1}\) is obtained by replacing the middle third of each side of \(I_{n}\) by a new outward equilateral triangle with sides of length \(1 / 3^{n+1}\). The limiting figure as \(n\ \rightarrow\ \infty\) is called the snowflake island. a. Let \(L_{n}\) be the perimeter of \(I_{n}\). Show that \(\lim_{n\rightarrow\infty}\ L_n=\infty\). b. Let \(A_{n}\) be the area of \(I_{n}\). Find \(\lim_{n\rightarrow\infty}\ A_n\). It exists!

41-46. Decimal expansions Write each repeating decimal first as a geometric series, then as a fraction (a ratio of two integers). 1.252525…

What is the condition for convergence of the geometric series \(\sum_{k=0}^{\infty} a r^{k}\)

19-34. Geometric series Evaluate the geometric series or state that it diverges. \(\sum_{k=0}^{\infty}\left(\frac{4}{3}\right)^{-k}\)

Multiplier effect Imagine that the government of a small community decides to give a total of $W, distributed equally, to all of its citizens. Suppose that each month each citizen saves a fraction p of his or her new wealth and spends the remaining 1 - p in the community. Assume no money leaves or enters the community, and all of the spent money is redistributed throughout the community. a. If this cycle of saving and spending continues for many months, how much money is ultimately spent? Specifically, by what factor is the initial investment of $W increased? (Economists refer to this increase in the investment as the multiplier effect.) b. Evaluate the limits \(p\ \rightarrow\ 0\) and \(p\ \rightarrow\ 1\) and interpret their meanings. (See Guided Projects for more on economic stimulus packages.)

19-34. Geometric series Evaluate the geometric series or state that it diverges. \(\sum_{k=0}^{\infty}\left(\frac{1}{4}\right)^{k} 5^{6-k}\)

Does a geometric sum always have a finite value?

Functions defined as series Suppose a function f is defined by the geometric series \(f(x)=\sum_{k=0}^{\infty}(-1)^k\ x^k\). a. Evaluate f(0), f(0.2), f(0.5), f(1), and f(1.5). b. What is the domain of f?

Geometric sums Evaluate the following geometric sums. \(\sum _{k=1}^{10}\ (\frac{4}{7})^k\)

47-58. Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\{S_{n}\right\}\). Then evaluate \(\lim_{n\rightarrow\infty}\ S_n\), to obtain the value of the series or state that the series diverges. \(\sum_{k=0}^{\infty} \frac{1}{16 k^{2}+8 k-3}\)

47-58. Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\{S_{n}\right\}\). Then evaluate \(\lim_{n\rightarrow\infty}\ S_n\), to obtain the value of the series or state that the series diverges. \(\sum_{k=1}^{\infty}\left(\frac{1}{k+2}-\frac{1}{k+3}\right)\)

35-40. Geometric series with alternating signs Evaluate the geometric series or state that it diverges. \(\sum_{k=2}^{\infty}(-0.15)^{k}\)

19-34. Geometric series Evaluate the geometric series or state that it diverges. \(\sum_{k=0}^{\infty} 0.9^{k}\)

What is meant by the ratio of a geometric series?

47-58. Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\{S_{n}\right\}\). Then evaluate \(\lim_{n\rightarrow\infty}\ S_n\), to obtain the value of the series or state that the series diverges. \(\sum_{k=1}^{\infty}\left[\tan ^{-1}(k+1)-\tan ^{-1} k\right]\)

Geometric sums Evaluate the following geometric sums. \(\sum_{k=4}^{12} 2^{k}\)

China's one-son policy In 1978, in an effort to reduce population growth, China instituted a policy that allows only one child per family. One unintended consequence has been that, because of a cultural bias toward sons, China now has many more young boys than girls. To solve this problem, some people have suggested replacing the one-child policy with a one-son policy: A family may have children until a boy is born. Suppose that the one-son policy were implemented and that natural birth rates remained the same (half boys and half girls). Using geometric series, compare the total number of children under the two policies.

19-34. Geometric series Evaluate the geometric series or state that it diverges. \(\sum_{k=0}^{\infty}\left(\frac{1}{4}\right)^{k}\)

35-40. Geometric series with alternating signs Evaluate the geometric series or state that it diverges. \(\sum_{k=1}^{\infty}(-e)^{-k}\)

47-58. Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\{S_{n}\right\}\). Then evaluate \(\lim_{n\rightarrow\infty}\ S_n\), to obtain the value of the series or state that the series diverges. \(\sum_{k=1}^{\infty} \frac{1}{(k+p)(k+p+1)}\), where p is a positive integer

What is the difference between a geometric sum and a geometric series?

35-40. Geometric series with alternating signs Evaluate the geometric series or state that it diverges. \(\sum_{k=1}^{\infty} 3\left(-\frac{1}{8}\right)^{3 k}\)

Geometric sums Evaluate the following geometric sums. \(\sum_{k=0}^{10}\left(\frac{1}{4}\right)^{k}\)

19-34. Geometric series Evaluate the geometric series or state that it diverges. \(\sum_{k=0}^{\infty} \frac{2^{k}}{7^{k}}\)

Geometric sums Evaluate the following geometric sums. \(\sum_{k=0}^{8} 3^{k}\)

47-58. Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\{S_{n}\right\}\). Then evaluate \(\lim_{n\rightarrow\infty}\ S_n\), to obtain the value of the series or state that the series diverges. \(\sum_{k=1}^{\infty}\left(\frac{1}{k+1}-\frac{1}{k+2}\right)\)

Functions defined as series Suppose a function f is defined by the geometric series \(f(x)=\sum_{k=0}^{\infty} x^{2 k}\) a. Evaluate f(0), f(0.2), f(0.5), f(1), and f(1.5). b. What is the domain of f?

47-58. Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\{S_{n}\right\}\). Then evaluate \(\lim_{n\rightarrow\infty}\ S_n\), to obtain the value of the series or state that the series diverges. \(\sum_{k=0}^{\infty}\left[\sin \left(\frac{(k+1) \pi}{2 k+1}\right)-\sin \left(\frac{k \pi}{2 k-1}\right)\right]\)

Geometric sums Evaluate the following geometric sums. \(\sum _{k=0}^6\ \pi^k\)

Periodic doses Suppose that you take 200 mg of an antibiotic every 6 hr. The half-life of the drug is 6 hr (the time it takes for half of the drug to be eliminated from your blood). If you continue this regimen indefinitely, what is the long-term (steady-state) amount of antibiotic in your blood?

House loan Suppose you take out a home mortgage for $180,000 at a monthly interest rate of 0.5%. If you make payments of $1000 per month, after how many months will the loan balance be zero? Estimate the answer by graphing the sequence of loan balances and then obtain an exact answer.

19-34. Geometric series Evaluate the geometric series or state that it diverges. \(\sum_{k=1}^{\infty} \frac{3^{k-1}}{4^{k+1}}\)

47-58. Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\{S_{n}\right\}\). Then evaluate \(\lim_{n\rightarrow\infty}\ S_n\), to obtain the value of the series or state that the series diverges. \(\sum_{k=1}^{\infty} \ln \left(\frac{k+1}{k}\right)\)

74-77. Comparing remainder terms Use Exercise 73 to determine how many terms of each series are needed so that the partial sum is within \(10^{-6}\) of the value of the series (that is, to ensure \(R_{n}<10^{-6}\)). a. \(\sum_{k=0}^{\infty}(-0.8)^{k}\) b. \(\sum_{k=0}^{\infty} 0.2^{k}\)

19-34. Geometric series Evaluate the geometric series or state that it diverges. \(\sum_{j=0}^{\infty}\left(\frac{1}{\pi}\right)^{j}\)

41-46. Decimal expansions Write each repeating decimal first as a geometric series, then as a fraction (a ratio of two integers). 0.121212…

Car loan Suppose you borrow $20,000 for a new car at a monthly interest rate of 0.75%. If you make payments of $600 per month, after how many months will the loan balance be zero? Estimate the answer by graphing the sequence of loan balances and then obtain an exact answer.

19-34. Geometric series Evaluate the geometric series or state that it diverges. \(\sum_{m=2}^{\infty} \frac{5}{2^{m}}\)

41-46. Decimal expansions Write each repeating decimal first as a geometric series, then as a fraction (a ratio of two integers). 1.00393939…

Geometric sums Evaluate the following geometric sums. \(\sum_{k=1}^{5}(-2.5)^{k}\)

19-34. Geometric series Evaluate the geometric series or state that it diverges. \(\sum_{k=1}^{\infty} 2^{-3 k}\)

41-46. Decimal expansions Write each repeating decimal first as a geometric series, then as a fraction (a ratio of two integers). 0.00952952…

7-18. Geometric sums Evaluate the following geometric sums. \(\frac{1}{3}+\frac{1}{5}+\frac{3}{25}+\frac{9}{125}+\cdots+\frac{243}{15,625}\)

41-46. Decimal expansions Write each repeating decimal first as a geometric series, then as a fraction (a ratio of two integers). 0.456456…

19-34. Geometric series Evaluate the geometric series or state that it diverges. \(\sum_{k=3}^{\infty} \frac{3 \cdot 4^{k}}{7^{k}}\)

35-40. Geometric series with alternating signs Evaluate the geometric series or state that it diverges. \(\sum_{k=1}^{\infty}\left(-\frac{2}{3}\right)^{k}\)

Series in an equation For what values of x does the geometric series \(f(x)=\sum_{k=0}^{\infty}\left(\frac{1}{1+x}\right)^{k}\) converge? Solve f(x) = 3.

Bouncing ball for time Suppose a rubber ball, when dropped from a given height, returns to a fraction p of that height. How long does it take for a ball dropped from 10 m to come to rest? In the absence of air resistance, a ball dropped from a height h requires \(\sqrt{2 h / g}\) seconds to fall to the ground, where \(g\approx9.8\mathrm{\ m}/\mathrm{s}^2\) is the acceleration due to gravity. The time taken to bounce up to a given height equals the time to fall from that height to the ground.

47-58. Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\{S_{n}\right\}\). Then evaluate \(\lim_{n\rightarrow\infty}\ S_n\), to obtain the value of the series or state that the series diverges. \(\sum_{k=0}^{\infty} \frac{1}{(3 k+1)(3 k+4)}\)

Values of a series a. Find the value for the series \(\sum_{k=1}^{\infty} \frac{3^{k}}{\left(3^{k+1}-1\right)\left(3^{k}-1\right)}\) b. For what value of a does the series \(\sum_{k=1}^{\infty} \frac{a^{k}}{\left(a^{k+1}-1\right)\left(a^{k}-1\right)}\) Coverage, and in those cases, what is its value?

Bubbles Imagine a stack of hemispherical soap bubbles with decreasing radii \(r_1=1,\ r_2,\ r_3,\ \ldots\) (see figure). Let \(h_{n}\) be the distance between the diameters of bubble n and bubble n + 1, and let \(H_{n}\) be the total height of the stack with n bubbles. a. Use the Pythagorean theorem to show that in a stack with n bubbles, \(h_1^2=r_1^2-r_2^2,\ h_2^2=r_2^2-r_3^2\), and so forth. Note that \(h_{n}=r_{n}\). b. Use part (a) to show that the height of a stack with n bubbles is \(H_{n}=\sqrt{r_{1}^{2}-r_{2}^{2}}+\sqrt{r_{2}^{2}-r_{3}^{2}}+\cdots\) \(+\sqrt{r_{n-1}^{2}-r_{n}^{2}}+r_{n}\). c. The height of a stack of bubbles depends on how the radii decrease. Suppose that \(r_1=1,\ r_2=a,\ r_3=a^2,\ldots,\ r_n=a^n\), where 0 < a < 1 is a fixed real number. In terms of a, find the height \(H_{n}\) of a stack with n bubbles. d. Suppose the stack in part (c) is extended indefinitely \((n\ \longrightarrow\ \infty)\). In terms of a, how high would the stack be? e. Challenge problem: Fix n and determine the sequence of radii, \(r_1,\ r_2,\ r_3,\ldots,\ r_n\) that maximizes \(H_{n}\), the height of the stack with n bubbles.

35-40. Geometric series with alternating signs Evaluate the geometric series or state that it diverges. \(3 \sum_{k=0}^{\infty} \frac{(-1)^{k}}{\pi^{k}}\)

Functions defined as series Suppose a function f is defined by the geometric series \(f(x)=\sum_{k=0}^{\infty} x^{k}\). a. Evaluate f(0), f(0.2), f(0.5), f(1), and f(1.5), if possible. b. What is the domain of f?

Remainder term Consider the geometric series \(S=\sum_{k=0}^{\infty} r^{k}\), which has the value 1/(1 - r) provided |r| < 1, and let \(S_{n}=\sum_{k=0}^{n-1} r^{k}=\frac{1-r^{n}}{1-r}\) be the sum of the first n terms. The remainder \(R_{n}\) is the error in approximating S by \(S_{n}\). Show that \(R_{n}=\left|S-S_{n}\right|=\left|\frac{r^{n}}{1-r}\right|\)

19-34. Geometric series Evaluate the geometric series or state that it diverges. \(\sum_{k=4}^{\infty} \frac{1}{5^{k}}\)

19-34. Geometric series Evaluate the geometric series or state that it diverges. \(\sum_{k=1}^{\infty} e^{-2 k}\)

Double glass An insulated window consists of two parallel panes of glass with a small spacing between them. Suppose that each pane reflects a fraction p of the incoming light and transmits the remaining light. Considering all reflections of light between the panes, what fraction of the incoming light is ultimately transmitted by the window? Assume the amount of incoming light is 1.

35-40. Geometric series with alternating signs Evaluate the geometric series or state that it diverges. \(\sum_{k=0}^{\infty}\left(-\frac{9}{10}\right)^{k}\)

47-58. Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\{S_{n}\right\}\). Then evaluate \(\lim_{n\rightarrow\infty}\ S_n\), to obtain the value of the series or state that the series diverges. \(\sum_{k=1}^{\infty}(\sqrt{k+1}-\sqrt{k})\)

47-58. Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\{S_{n}\right\}\). Then evaluate \(\lim_{n\rightarrow\infty}\ S_n\), to obtain the value of the series or state that the series diverges. \(\sum_{k=1}^{\infty}\left(\frac{1}{\sqrt{k+1}}-\frac{1}{\sqrt{k+3}}\right)\)

74-77. Comparing remainder terms Use Exercise 73 to determine how many terms of each series are needed so that the partial sum is within \(10^{-6}\) of the value of the series (that is, to ensure \(R_{n}<10^{-6}\)). a. \(\sum_{k=0}^{\infty} 0.6^{k}\) b. \(\sum_{k=0}^{\infty} 0.15^{k}\)

19-34. Geometric series Evaluate the geometric series or state that it diverges. \(\sum_{k=0}^{\infty}\left(\frac{e}{\pi}\right)^{k}\)

Does a geometric series always have a finite value?

Geometric sums Evaluate the following geometric sums. \(\sum_{k=0}^{20}\left(\frac{2}{5}\right)^{2 k}\)

47-58. Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\{S_{n}\right\}\). Then evaluate \(\lim_{n\rightarrow\infty}\ S_n\), to obtain the value of the series or state that the series diverges. \(\sum_{k=1}^{\infty} \frac{1}{(a k+1)(a k+a+1)}\), where a is a positive integer

19-34. Geometric series Evaluate the geometric series or state that it diverges. \(\sum_{k=0}^{\infty}\left(\frac{3}{5}\right)^{k}\)

7-18. Geometric sums Evaluate the following geometric sums. \(1+\frac{2}{3}+\frac{4}{9}+\frac{8}{27}\)

What is the defining characteristic of a geometric series? Give an example.

74-77. Comparing remainder terms Use Exercise 73 to determine how many terms of each series are needed so that the partial sum is within \(10^{-6}\) of the value of the series (that is, to ensure \(R_{n}<10^{-6}\)). a. \(\sum_{k=0}^{\infty} 0.72^{k}\) b. \(\sum_{k=0}^{\infty}(-0.25)^{k}\)

Fish harvesting A fishery manager knows that her fish population naturally increases at a rate of 1.5% per month. At the end of each month, 120 fish are harvested. Let \(F_{n}\) be the fish population after the nth month, where \(F_{0}=4000\) fish. Assuming that this process continues indefinitely, what is the long-term (steady-state) population of the fish?

41-46. Decimal expansions Write each repeating decimal first as a geometric series, then as a fraction (a ratio of two integers). 5.12838383…

74-77. Comparing remainder terms Use Exercise 73 to determine how many terms of each series are needed so that the partial sum is within \(10^{-6}\) of the value of the series (that is, to ensure \(R_{n}<10^{-6}\)). a. \(\sum_{k=0}^{\infty}\left(\frac{1}{\pi}\right)^{k}\) b. \(\sum_{k=0}^{\infty}\left(\frac{1}{e}\right)^{k}\)

47-58. Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\{S_{n}\right\}\). Then evaluate \(\lim_{n\rightarrow\infty}\ S_n\), to obtain the value of the series or state that the series diverges. \(\sum_{k=1}^{\infty} \frac{1}{(k+1)(k+2)}\)

19-34. Geometric series Evaluate the geometric series or state that it diverges. \(\sum_{k=0}^{\infty} 1.01^{k}\)

Decimal expansions a. Consider the number 0.555555..., which can be viewed as the series \(5 \sum_{k=1}^{\infty} 10^{-k}\). Evaluate the geometric series to obtain a rational value of 0.555555… b. Consider the number 0.54545454..., which can be represented by the series \(54 \sum_{k=1}^{\infty} 10^{-2 k}\). Evaluate the geometric series to obtain a rational value of the number. c. Now generalize parts (a) and (b). Suppose you are given a number with a decimal expansion that repeats in cycles of length p, say, \(n_1,\ n_2\ldots n_p,\) where \(n_1,\ldots,\ n_p\) are integers between 0 and 9. Explain how to use geometric series to obtain a rational form of the number. d. Try the method of part (c) on the number 0.123456789123456789.... e. Prove that \(0 . \overline{9}=1\).

19-34. Geometric series Evaluate the geometric series or state that it diverges. \(\sum_{k=2}^{\infty}\left(\frac{3}{8}\right)^{3 k}\)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. \(\sum_{k=1}^{\infty}\left(\frac{\pi}{e}\right)^{-k}\) is a convergent geometric series. b. If a is a real number and \(\sum_{k=12}^{\infty} a^{k}\) converges, then \(\sum_{k=1}^{\infty} a^{k}\) converges. c. If the series \(\sum_{k=1}^{\infty} a^{k}\) converges and |a| < |b|, then the series \(\sum_{k=1}^{\infty} b^{k}\) converges.

Zeno's paradox The Greek philosopher Zeno of Elea (who lived about 450 B.C.) invented many paradoxes, the most famous of which tells of a race between the swift warrior Achilles and a tortoise. Zeno argued The slower when running will never be overtaken by the quicker; for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead. In other words, giving the tortoise a head start, Achilles will never overtake the tortoise because every time Achilles reaches the point where the tortoise was, the tortoise has moved ahead. Resolve this paradox by assuming that Achilles gives the tortoise a 1-mi head start and runs 5 mi/hr to the tortoise's 1 mi/hr. How far does Achilles run before he overtakes the tortoise, and how long does it take?

Archimedes' quadrature of the parabola The Greeks solved several calculus problems almost 2000 years before the discovery of calculus. One example is Archimedes' calculation of the area of the region R bounded by a segment of a parabola, which he did using the “method of exhaustion.” As shown in the figure, the idea was to fill R with an infinite sequence of triangles. Archimedes began with one triangle inscribed in the parabola, with area \(A_{1}\), and proceeded in stages, with the number of new triangles doubling at each stage. He was able to show (the key to the solution) that at each stage, the area of a new triangle is \(\frac{1}{8}\) of the area of a triangle at the previous stage; for example, \(A_{2}=\frac{1}{8} A_{1}\), and so forth. Show, as Archimedes did, that the area of R is \(\frac{4}{3}\) times the area of \(A_{1}\).