#76 part a 8:32ll1 Safari< RealAnalysis-ISBN-fix...is just a convenient notational shorthand telling us that the sequence divergesby becoming arbitrarily large.Problem 76. Suppose lim an = 0 and lim b, = -0 and a e R. Prove orgive a counterezample:CONVERGENCE OF SEQUENCES AND SERIES86(a) lim a, + b, = 00(b) lim a,b, = -00(c) lim aa, = 00(d) lim ab, = -00Finally, a sequence can diverge in other ways as the following problem dis-plays.Problem 77. Show that each of the following sequences diverge.(a) ((-1)")(b) (-1)" п)я-1(1if n = 2P for some peN(e) an =! otherwise,Problem 78. Suppose that (a,) diverges but not to infinity and that a is areal number. What conditions on a will guarantee that:(a) (aa,) converges?(b) (aa,) diverges?Problem 79. Show that if r| > 1 then (r") diverges. Will it diverge toinfinity?Additional ProblemsProblem 80. Prove that if lim,- Sn = 8 then lim- |Sn| = |s|. Prove thatthe converse is true uhen s = 0, but it is not necessarily true otherwise.Problem 81.(a) Let (sn) and (tn) be sequences with s, < tn, Vn. Suppose lim,- 8n = sand lim,- t, = t. Prove s < t. [Hint: Assume for contradiction, that8>t and use the definition of converaence with e = to produce an nNext1DashboardCalendarTo DoNotificationsInbox因

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College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter9: Sequences, Probability And Counting Theory

Section9.1: Sequences And Their Notations

Problem 66SE: Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a Tl-84, do...

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Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a Tl-84, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose ‘seq(” from the dropdown list. Press [ENTER] • In the line headed “Expr:” type in the explicit formula, using the [X,T,,n] button for n • In the line headed ‘Variable” type In the variable used on the previous step. • In the line headed ‘start:” key in the value of n that begins the sequence. • In the line headed “end:’ key in the value of n that ends the sequence. • Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Using a TI-83, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose “seq(“ from the dropdown list. Press [ENTER]. • Enter the items in the order “Expr’, Variable’, ‘start”. end separated by commas. See the instructions above for the description of each item. • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary. 65. List the first four terms of the sequence. an=5.7n+0.275(n1)
Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a Tl-84, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose ‘seq(” from the dropdown list. Press [ENTER] • In the line headed “Expr:” type in the explicit formula, using the [X,T,,n] button for n • In the line headed ‘Variable” type In the variable used on the previous step. • In the line headed ‘start:” key in the value of n that begins the sequence. • In the line headed “end:’ key in the value of n that ends the sequence. • Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Using a TI-83, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose “seq(“ from the dropdown list. Press [ENTER]. • Enter the items in the order “Expr’, Variable’, ‘start”. end separated by commas. See the instructions above for the description of each item. • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary. 66. List the first six terms of the sequence an=n!n
Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a Tl-84, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose ‘seq(” from the dropdown list. Press [ENTER] • In the line headed “Expr:” type in the explicit formula, using the [X,T,,n] button for n• In the line headed ‘Variable” type In the variable used on the previous step. • In the line headed ‘start:” key in the value of n that begins the sequence. • In the line headed “end:’ key in the value of nthat ends the sequence. • Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Using a TI-83, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose “seq(“ from the dropdown list. Press [ENTER]. • Enter the items in the order “Expr’, Variable’, ‘start”. end separated by commas. See the instructions above for the description of each item. • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary. 63. List the first six terms of the sequence. an=n33.5n2+4.1n1.52.4n

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is just a convenient notational shorthand telling us that the sequence diverges by becoming arbitrarily large. Problem 76. Suppose lim a, = o and lim bn = -o and a €R. Prove or give a counterezample: CONVERGENCE OF SEQUENCES AND SERIES 86 (a) lim a, + b, = 00