Comparing Exponential and Faetorial (irowtit Consiilei the sequence (( = l(l"/!.(a) Find

In Exercises 1-12, write tlie first five terms of the sequence.

In Exercises 1-12, write tlie first five terms of the sequence. " ~ /( + 3

In Exercises 1-12, write tlie first five terms of the sequence. , = HT

In Exercises 1-12, write tlie first five terms of the sequence. ^'ii = i-W

In Exercises 1-12, write tlie first five terms of the sequence. = . IITT Sin

In Exercises 1-12, write tlie first five terms of the sequence. ti,, = IITT

In Exercises 1-12, write tlie first five terms of the sequence. = (-1) ^"''

In Exercises 1-12, write tlie first five terms of the sequence. <' = (-1)"^'(-' \/i/

In Exercises 1-12, write tlie first five terms of the sequence. = 1 1 n ir

In Exercises 1-12, write tlie first five terms of the sequence. n ir "" = lO + ^ + A )i II I

In Exercises 1-12, write tlie first five terms of the sequence. = 3"

In Exercises 1-12, write tlie first five terms of the sequence. " = :vi'. in - 1)!

In Exercises 13-16, write the first five terms of the recursively defined sequence, c?! = 3, (ij^i = 2(t/, - 1)

In Exercises 13-16, write the first five terms of the recursively defined sequence, (/, = 4, i/j ,

In Exercises 13-16, write the first five terms of the recursively defined sequence, a, = 32. (((^ I = 3(7(

In Exercises 13-16, write the first five terms of the recursively defined sequence, </, = 6, 0^ + , = \a^

In Exercises 17-20, match the sequence with its graph, [The graphs are ialu'led (a), (h), (c), and (d).|

In Exercises 17-20, match the sequence with its graph, [The graphs are ialu'led (a), (h), (c), and (d).|

In Exercises 17-20, match the sequence with its graph, [The graphs are ialu'led (a), (h), (c), and (d).|

In Exercises 17-20, match the sequence with its graph, [The graphs are ialu'led (a), (h), (c), and (d).|

In E^xcrcises 21-26, use a graphing utility to graph the first ten terms of the sequence. a

In E^xcrcises 21-26, use a graphing utility to graph the first ten terms of the sequence. a = 2 - 11

In E^xcrcises 21-26, use a graphing utility to graph the first ten terms of the sequence. a = 16(-0.5)"-'

In E^xcrcises 21-26, use a graphing utility to graph the first ten terms of the sequence. (/ = 8(0.75)"

In E^xcrcises 21-26, use a graphing utility to graph the first ten terms of the sequence. a 2n

In E^xcrcises 21-26, use a graphing utility to graph the first ten terms of the sequence. - -"'

In Exercises 27-30, write tlie next two apparent terms of tlie sequence, Descrilje the pattern you used to find these terms. 2.?, 8, II. .

In Exercises 27-30, write tlie next two apparent terms of tlie sequence, Descrilje the pattern you used to find these terms. i. 4, 5. 5. . . .

In Exercises 27-30, write tlie next two apparent terms of tlie sequence, Descrilje the pattern you used to find these terms. 3,"ii-i

In Exercises 27-30, write tlie next two apparent terms of tlie sequence, Descrilje the pattern you used to find these terms. 5. 10.20.40. .

In Exercises 31-36, simplify the ratio of factorials.

In Exercises 31-36, simplify the ratio of factorials.

In Exercises 31-36, simplify the ratio of factorials.

In Exercises 31-36, simplify the ratio of factorials.

In Exercises 31-36, simplify the ratio of factorials.

In Exercises 31-36, simplify the ratio of factorials.

In Exercises 37-42, find the limit (if possible) of the sequence,

In Exercises 37-42, find the limit (if possible) of the sequence,

In Exercises 37-42, find the limit (if possible) of the sequence,

In Exercises 37-42, find the limit (if possible) of the sequence,

In Exercises 37-42, find the limit (if possible) of the sequence,

In Exercises 37-42, find the limit (if possible) of the sequence,

In Exercises 43-46, use a graphing utility to graph the first ten terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your infer ence analytically and. if the sequence converges, find its limit.

In Exercises 43-46, use a graphing utility to graph the first ten terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your infer ence analytically and. if the sequence converges, find its limit.

In Exercises 43-46, use a graphing utility to graph the first ten terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your infer ence analytically and. if the sequence converges, find its limit.

In Exercises 43-46, use a graphing utility to graph the first ten terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your infer ence analytically and. if the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 47-66, determine the convergence or divergence of the sequence with the given ;;th term. If the sequence converges, find its limit.

In Exercises 67-S(), vvrilc an expression for the tli term of the sequence. (There is more than one correct answer.) 1.4.7, 10. . . .

In Exercises 67-S(), vvrilc an expression for the tli term of the sequence. (There is more than one correct answer.) 3. 7. II. KS, .

In Exercises 67-S(), vvrilc an expression for the tli term of the sequence. (There is more than one correct answer.) -1.2.7. 14.2.1. . . .

In Exercises 67-S(), vvrilc an expression for the tli term of the sequence. (There is more than one correct answer.) 1 _i 1 _-L

In Exercises 67-S(), vvrilc an expression for the tli term of the sequence. (There is more than one correct answer.) = i i i

In Exercises 67-S(), vvrilc an expression for the tli term of the sequence. (There is more than one correct answer.) i i i il I ' > s- s. 1 1- >

In Exercises 67-S(), vvrilc an expression for the tli term of the sequence. (There is more than one correct answer.) _. 1.1. 4, s-

In Exercises 67-S(), vvrilc an expression for the tli term of the sequence. (There is more than one correct answer.) _i 1 _i !* -^21 l-i. 5. 1. 4. x- 16-

In Exercises 67-S(), vvrilc an expression for the tli term of the sequence. (There is more than one correct answer.) 2, 1 + i 1 + 5, 1 + J, 1 + i . . .

In Exercises 67-S(), vvrilc an expression for the tli term of the sequence. (There is more than one correct answer.) I + 5. 1 + i 1 + |. 1 + 1^. 1 + ^. .

In Exercises 67-S(), vvrilc an expression for the tli term of the sequence. (There is more than one correct answer.) 2 3' 3 4' 4 ."^^ -"^ 6"

In Exercises 67-S(), vvrilc an expression for the tli term of the sequence. (There is more than one correct answer.) : (> :4' i2(i'

In Exercises 67-S(), vvrilc an expression for the tli term of the sequence. (There is more than one correct answer.) 3 .S 1 3 ."i 7

In Exercises 67-S(), vvrilc an expression for the tli term of the sequence. (There is more than one correct answer.) l.A,-. .V- .V .V X 6 ' 24' 12()'

In Exercises 81-90. determine whether the sequence with the given /;th term is monotonic. Discuss the boundedness of the sequence. Use a graphing utilit\ to c(mfirni \our results. (/ = 4 - I

In Exercises 81-90. determine whether the sequence with the given /;th term is monotonic. Discuss the boundedness of the sequence. Use a graphing utilit\ to c(mfirni \our results. </ 3/1 )( + :

In Exercises 81-90. determine whether the sequence with the given /;th term is monotonic. Discuss the boundedness of the sequence. Use a graphing utilit\ to c(mfirni \our results. " = ^

In Exercises 81-90. determine whether the sequence with the given /;th term is monotonic. Discuss the boundedness of the sequence. Use a graphing utilit\ to c(mfirni \our results. (i = lie

In Exercises 81-90. determine whether the sequence with the given /;th term is monotonic. Discuss the boundedness of the sequence. Use a graphing utilit\ to c(mfirni \our results. a = (- 1 r

In Exercises 81-90. determine whether the sequence with the given /;th term is monotonic. Discuss the boundedness of the sequence. Use a graphing utilit\ to c(mfirni \our results. CI.

In Exercises 81-90. determine whether the sequence with the given /;th term is monotonic. Discuss the boundedness of the sequence. Use a graphing utilit\ to c(mfirni \our results. <(

In Exercises 81-90. determine whether the sequence with the given /;th term is monotonic. Discuss the boundedness of the sequence. Use a graphing utilit\ to c(mfirni \our results. <' = 1^^

In Exercises 81-90. determine whether the sequence with the given /;th term is monotonic. Discuss the boundedness of the sequence. Use a graphing utilit\ to c(mfirni \our results. (/ = sin '

In Exercises 81-90. determine whether the sequence with the given /;th term is monotonic. Discuss the boundedness of the sequence. Use a graphing utilit\ to c(mfirni \our results. a =

In Exercises 91-94, (a) use Theorem 8.5 to show that the sequence with the given Hlh term converges and (h) use a graph ing utility to graph the tlrst ten terms of the sequence and find its limit. ci=5+

In Exercises 91-94, (a) use Theorem 8.5 to show that the sequence with the given Hlh term converges and (h) use a graph ing utility to graph the tlrst ten terms of the sequence and find its limit. = 4

In Exercises 91-94, (a) use Theorem 8.5 to show that the sequence with the given Hlh term converges and (h) use a graph ing utility to graph the tlrst ten terms of the sequence and find its limit. - = 3(1

In Exercises 91-94, (a) use Theorem 8.5 to show that the sequence with the given Hlh term converges and (h) use a graph ing utility to graph the tlrst ten terms of the sequence and find its limit. </ = 4 +

Coinpoiiiid Interest Consider the sequence |/4| whose ;ilh term is aivcn hv Pi 1; wiiere P is tlie principal. ,\ is tlie accounl balance alter n months, and ; is the interest rate compounded aniuially. (a) Is |.4] a con\'ert!ent sequence'.' Explain. (bl Find the tlrst ten terms ol the sequence il P = $'-)()IIO and ) = II. M.S.

Investnieiit A deposit of .Slot) is made at the beginning of each month in an account at an annual interest rate o! 12% compounded monthl\ The balance in the account alter 11 months is A = l(l()(mil[ll (III" - 1], (a) Compute the first si\ terms ol the scqticncc 1/4|. (bl find the balance after .^ \c.irs b\ computing tlie fiOili term of the sequence. (c) Fiiitl the balance after 20 years by conipiiting the 240th term of the sciiiicnce.

In your own words, define each ol the following, (a) Sequence (h) Convergence of a sequence (c) Bounded monotonic sc(.|ucnce

The giaplis of two sequences are given in the figures, i Which graph represents the sequence with alternating signs? Explain

In Exercises 99-102, give an example of a sequence satisfy ing the condition or explain why no such sequence exists. (Examples are not unique. I A nionotonically increasing sequence that converges to 1(3

In Exercises 99-102, give an example of a sequence satisfy ing the condition or explain why no such sequence exists. (Examples are not unique. I A monotonically increasing bounded sequence that does not converge

In Exercises 99-102, give an example of a sequence satisfy ing the condition or explain why no such sequence exists. (Examples are not unique. I A sequence that converges to J

In Exercises 99-102, give an example of a sequence satisfy ing the condition or explain why no such sequence exists. (Examples are not unique. I An unbounded sequence that converges to 1 00

Government Expendilnres A goxeinnicnt program llial currently costs taxpayers $2? billion per year is cut hack hy 20 percent per year. (a) Write an expression lor the amount hudgetcd h>r Ihis program after n years. (h) Compute the budgets for llie first 4 years. (c) Determine the convergence or diveigence ol the sequence of reduced budgets. If the sequence converges, find its limit.

Inflation If the rate of intlation is 4^% per year and the average price of a ear is ciiirently $16.0(1(1, the average price after ii years i-, P = SI 6,000(1.04.^1". Compute the ,i\ciage price lor the next 5 years.

Modelint; Ikilti The average cost per day for a hospital room from 1990 through 1997 is shown in the table, where </ is the average cost in dollars and n is the year, with ;; = con'e sponding to 1990. (Sauivc: American Hosjvhil Associiilinii} (a) Use the regression capabilities of a graphing utility to find a model of the form (( = Ivr + III + </, // = 0, 1, 2, 3, 4, 3, 6, 7 for the data. Use the graphing utility to plot the points and graph the model, (bl Use ihe model to predict the cost in the \car 2(104.

Modelinti Data The annual sales (/ (in millions of dollars) of H, .1. Hcin/ Company trom 1990 through 1999 are given below as ordered pairs ot the lonii in. a^). where ;/ is the year, with n = corresponding to 1990, iSinnvc: IW9 H. J. Heinz Report) (0,6086), (1.6647). (2, 6.SS2), (3,7103), (4,7047), (5,8087), (6,9112), (7,9357), (8,9209). (9, 9.i()()) (a) Use the regression capabilities of a graphing utility to find a model of the firm a = hii + c. )j = 0, 1 9 for the data. Graphically coinpare the points and the iholIcI. (b) LIse the model to predict sales in the year 2004.

Comparing Exponential and Faetorial (irowtit Consiilei the sequence (( = l(l"/!. (a) Find two consecutive terms that are e(.|ual in magnitude. (b) Are the terms following those found in part (a) increasing or decreasing? (c) In Section 7.7, E.xercises 63-68, it was shown that for "large" values of the independent variable an exponential function increases more rapidly than a polynomial function. From the resiili m |iarl (b), what inference can you make about (he rate of grow th of an exponential function versus a factorial function for "large" integer values of ?

Compute the first six terms of the sequence U/| = 1(1 + i///)"l. If the sequence converges, find its limit.

Compute the first six terms of the sequence U',,! = ) Z^n \. If the sequence converges, find its limit.

Prove that if j.v,J converges to L and L > 0, then there exists a luunber N such that .v > for ii > N.

ihonaeei Seqnenee In a study of the progeny of rabbits, Fibonacci (ca. 1 175-ca. 1250) encountered the sequence now hearing his name. It is defined recursively by where 1. (;, = 1 and -, (a) Write the first 12 terms of the sequence. (b) Write the first ten terms of the sequence defined by /'. . = n > \. (c) Using the definition m part (b), show that I /> 1 + (d) The golden ratio /) can be defined by lim /' = p. Show that ""'" P=\ + \/p and solve this equation for p.

Complete the proof of Theorem 8.5.

True or False? In Exercises 113-116, determine whether the .statement is true or false. If it is false, explain why or give an example that shows it is false. If |ii| conxerges to 3 and l/il converges to 2, then ]<(,, + /)} converges to 5.

True or False? In Exercises 113-116, determine whether the .statement is true or false. If it is false, explain why or give an example that shows it is false. If U/} converges, then Inn (i/ - </^|) = 0.

True or False? In Exercises 113-116, determine whether the .statement is true or false. If it is false, explain why or give an example that shows it is false. If;; > I, then ;;! = ;;(;; - I)!.

True or False? In Exercises 113-116, determine whether the .statement is true or false. If it is false, explain why or give an example that shows it is false. If 1(;| converges, then \ii,Jii\ converges to 0.

Consider the sequence J2. ^2 + 72. + V2 + s/2. . . . where o = 72 + fl _ , for ;; > 2. Compute the first five terms of this sequence. Find lim <;.

Conjecture Let .v,, = I and consider the sequence -v given bv the formula I I 1,2. Use a graphing utility to compute the first ten terms of the sequence and make a conjecture about the limit of the .sequence.