9/24/13 Section 8.1: Sequences
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1 WebAssig Sectio 8.1: Sequeces (Homework) Curret Score : 21 / 65 Due : Friday, October :05 PM CDT Jey Griffi Calculus 2, sectio Sectio 400, Fall 2013 Istructor: Richard Madrid 1. 5/5 poits Previous Aswers SCalc AE.002. Video Example EXAMPLE 2 Fid a formula for the geeral term of the sequece below, assumig that the patter of the first few terms cotiues. 2/3, -3/9, 4/27, -5/81, 6/243,... SOLUTION We are give that a 1, a 2 -, a 3, a 4 -, a tice that the umerators of these fractios start with 2 ad icrease by wheever we go to the ext term. The secod term has umerator 3, the third term has umerator 4; i geeral, the th term will have umerator. The deomiators are powers of 3, so a has deomiator. The sigs of the terms are alterately positive ad egative so we eed to multiply by a power of -1. Here we wat to start with a positive term ad so we use (-1) -1 or (-1) +1. Therefore, (-1) /8
2 2. 3/3 poits Previous Aswers SCalc AE.010. Video Example EXAMPLE 9 Discuss the covergece of the sequece below, where! ! 2 SOLUTION Both umerator ad deomiator approach ifiity as, but here we have o correspodig fuctio for use with l'hospital's rule (x! is ot defied whe x is ot a iteger.) Let's write out a few terms to get a feelig for what happes to a as gets large. a 1 3 3(1 2) a (1 2 3) a ( ) It appears from these expressios that the terms are decreasig ad perhaps approach 0. To cofirm this, observe that ( ) tice that the expressio i paretheses is at most 1 because the umerator is less tha (or equal to) the deomiator. So 0 < a < We kow that 3/ 2 0 as. Therefore a 0 as by the Squeeze Theorem. 2/8
3 3. 6/6 poits Previous Aswers SCalc AE.011. EXAMPLE 10 For what values of r is the sequece below coverget? (r /4)^(2 ) SOLUTION We kow that x a x for a > 1 ad x a x 0 for 0 < a < 1. Therefore, puttig a r/4 ad usig Theorem 3, we have _(->ifiity) (r /4)^(2 ) {(ifiity text( if ) r /4 > 1,0 text( if ) 0 < r /4 < 1) It is obvious that _(->ifiity)1^(2 ) 1 _(->ifiity)0^(2 ) ad If -1 < r/4 < 0, the 0 < r/4 < 1, so _(->ifiity) (r /4)^(2 ) _(->ifiity) r /4 ^(2 ) 0 Video Example ad therefore x (r /4) 2 0 by Theorem 6. If r -4 the {(r/4) 2 }. The figures show the graphs for various values of r. 4. 5/5 poits Previous Aswers TaCalcET Write the first five terms of the sequece a whose th term is give. a 2 a 3 a 4 a 5 5/3 6/5 1 8/9 5. 2/5 poits Previous Aswers TaCalcET Write the first five terms of the sequece whose th term is give. a 1 a 2 a 3 a 4 1 3/2 5/6 7/ (2 1)! a 5 9/ /8
4 6. /5 poits TaCalcET Write the first five terms of the sequece a 1 3, a + 1 3a + 1 {a} whose th term is give. a 1 a 2 a 3 a 4 a 5 7. /1 poits TaCalcET Fid a expressio for the th term of the sequece. (Assume that the patter cotiues.) ,,,,, /1 poits TaCalcET Fid a expressio for the th term of the sequece. (Assume that the patter cotiues.) ,,,,, /2 poits TaCalcET or. If it, fid its it. (If the quatity, eter DIVERGES.) 10. /2 poits TaCalcET or ( 1) If it, fid its it. (If the quatity, eter DIVERGES.) 4/8
5 11. /2 poits TaCalcET ( 1) 3/2 or. If it, fid its it. (If the quatity, eter DIVERGES.) 12. /2 poits TaCalcET or. ( 1) If it, fid its it. (If the quatity, eter DIVERGES.) 13. /2 poits TaCalcET or. 6 + ( 1) If it, fid its it. (If the quatity, eter DIVERGES.) 14. /2 poits TaCalcET or. If it, fid its it. (If the quatity, eter DIVERGES.) 5/8
6 15. /2 poits TaCalcET /2 + 1/ /3 or. If it, fid its it. (If the quatity, eter DIVERGES.) 16. /2 poits TaCalcET si π or. If it, fid its it. (If the quatity, eter DIVERGES.) 17. /2 poits TaCalcET ta 1 ( 2 ) or. If it, fid its it. (If the quatity, eter DIVERGES.) 18. /2 poits TaCalcET or e If it, fid its it. (If the quatity, eter DIVERGES.) a 6/8
7 19. /2 poits TaCalcET or. + 7 If it, fid its it. (If the quatity, eter DIVERGES.) 20. /2 poits TaCalcET / or. If it, fid its it. (If the quatity, eter DIVERGES.) 21. /2 poits TaCalcET ( 4)! or. If it, fid its it. (If the quatity, eter DIVERGES.) 22. /1 poits TaCalcET Evaluate. Hit: Use Theorem /8
8 23. /2 poits TaCalcET ( 1) is mootoic. Is the sequece bouded? 24. /2 poits TaCalcET si is mootoic. Is the sequece bouded? 25. /2 poits TaCalcET l( + 2) + 2 is mootoic. Is the sequece bouded? 26. /1 poits TaCalcET Fid the it of the sequece,,, Hit: Show that 6 (2 1)/ /2. 8/8
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