Do the following scquences {a„} converge or diverge? Justify your answers. (a) an = 3 (b) an – (-1)"/n (c) a, = (-1)"- dm – (-1)" (d) an = (-1)an+1 ( 1)" i n 1)" + (e) an - ( 1)» -n In (e*)") (f) an 3n (8) 4n - (-1)* cos ((n+1)) (h) an – (-1)" sin (5 (2n + 1))

Do the following scquences {a„} converge or diverge? Justify your answers. (a) an = 3 (b) an – (-1)"/n (c) a, = (-1)"- dm – (-1)" (d) an = (-1)an+1 ( 1)" i n 1)" + (e) an - ( 1)» -n In (e)") (f) an 3n (8) 4n - (-1) cos ((n+1)) (h) an – (-1)" sin (5 (2n + 1))

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.3: Geometric Sequences

Problem 82E

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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
Suppose that an object starts with an initial velocity of v0(in ft/sec) and moves under a constant acceleration a (in ft/sec2). Then the velocity vn (in ft/sec) after n seconds is given by vn = v0 + an. Show that this sequence is arithmetic. (Hint: Show that vn+1 − vn is constant.)
Show that the sequence whose nth term is an = n2/ (2n − 1) converges.
A sequence that does not have any convergent subsequence
A) Find Sn of this sequence B) is Sn convergent or divergent (Sn is the sum of all the terms of this sequence)
i. Suppose that the sn satisfies both limn→∞ s2n = 3 and limn→∞ s2n+1 = 3. (That is, the sequence given by the even terms of sn and that given by the odd terms of sn both converge to 3.) Show that also limn→∞ sn = 3.ii. Give an example of a sequence where the sequences given by the even and by the odd terms both converge, but where the entire sequence does not converge.
i. Suppose that the sn satisfies both limn→∞s2n = 3 and limn→∞s2n+1 = 3. (That is, the sequence given by the even terms of sn and that given by the odd terms of sn both converge to 3.) Show that also limn→∞sn = 3.ii. Give an example of a sequence where the sequences given by the even and by the odd terms both converge, but where the entire sequence does not converge.
Show that the sequence {Sn} give in general terms is convergent.
E. i. Suppose that the sn satisfies both limn→∞ s2n = 3 and limn→∞ s2n+1 = 3. (That is, the sequence given by the even terms of sn and that given by the odd terms of sn both converge to 3.) Show that also limn→∞ sn = 3.ii. Give an example of a sequence where the sequences given by the even and by the odd terms both converge, but where the entire sequence does not converge. .
Choose the true statements about a convergent series.