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1 Preface Here are the solutios to the practice problems for my otes. Some solutios will have more or less detail tha other solutios. As the difficulty level of the problems icreases less detail will go ito the basics of the solutio uder the assumptio that if you ve reached the level of workig the harder problems the you will probably already uderstad the basics fairly well ad wo t eed all the explaatio. This documet was writte with presetatio o the web i mid. O the web most solutios are broke dow ito steps ad may of the steps have hits. Each hit o the web is give as a popup however i this documet they are listed prior to each step. Also, o the web each step ca be viewed idividually by clickig o liks while i this documet they are all showig. Also, there are liable to be some formattig parts i this documet iteded for help i geeratig the web pages that have t bee removed here. These issues may make the solutios a little difficult to follow at times, but they should still be readable. Sequeces. List the first 5 terms of the followig sequece. 7 = 0 Solutio There really is t all that much to this problem. All we eed to do is, startig at = 0, plug i the first five values of ito the formula for the sequece terms. Doig that gives, 0 = 0: = = : = : 7 = : = = = : = 7 9 So, the first five terms of the sequece are, 007 Paul Dawkis

2 8 6 0,,, 6,, 9 Note that we put the formal aswer iside the braces to make sure that we do t forget that we are dealig with a sequece ad we made sure ad icluded the at the ed to remider ourselves that there are more terms to this sequece that just the five that we listed out here.. List the first 5 terms of the followig sequece. + ( ) + = Solutio There really is t all that much to this problem. All we eed to do is, startig at =, plug i the first five values of ito the formula for the sequece terms. Doig that gives, So, the first five terms of the sequece are, = = : = : = 5: 5 5 = 6: ,,,,, 89 7 Note that we put the formal aswer iside the braces to make sure that we do t forget that we are dealig with a sequece ad we made sure ad icluded the at the ed to remider ourselves that there are more terms to this sequece that just the five that we listed out here. 007 Paul Dawkis

3 . Determie if the give sequece coverges or diverges. If it coverges what is its limit? = Step To aswer this all we eed is the followig limit of the sequece terms. 7+ lim = 0 + You do recall how to take limits at ifiity right? If ot you should go back ito the Calculus I material do some refreshig o limits at ifiity as well at L Hospital s rule. Step We ca see that the limit of the terms existed ad was a fiite umber ad so we kow that the sequece coverges ad its limit is.. Determie if the give sequece coverges or diverges. If it coverges what is its limit? ( ) + Step To aswer this all we eed is the followig limit of the sequece terms. lim ( ) = 0 + However, because of the we ca t compute this limit usig our kowledge of computig limits from Calculus I. Step Recall however, that we had a ice Fact i the otes from this sectio that had us computig ot the limit above but istead computig the limit of the absolute value of the sequece terms. ( ) lim = lim = This is a limit that we ca compute because the absolute value got rid of the alteratig sig, i.e. the Paul Dawkis

4 Step Now, by the Fact from class we kow that because the limit of the absolute value of the sequece terms was zero (ad recall that to use that fact the limit MUST be zero!) we also kow the followig limit. ( ) lim = 0 + Step We ca see that the limit of the terms existed ad was a fiite umber ad so we kow that the sequece coverges ad its limit is zero. 5. Determie if the give sequece coverges or diverges. If it coverges what is its limit? e 5 e = Step To aswer this all we eed is the followig limit of the sequece terms. 5 5 e 5 5 lim lim lim = e = e e e = You do recall how to use L Hospital s rule to compute limits at ifiity right? If ot you should go back ito the Calculus I material do some refreshig. Step We ca see that the limit of the terms existed ad but was ifiite ad so we kow that the sequece diverges. 6. Determie if the give sequece coverges or diverges. If it coverges what is its limit? ( + ) ( + ) l l Step To aswer this all we eed is the followig limit of the sequece terms. = l ( + ) lim lim + = + = lim = l ( + ) ( + ) Paul Dawkis

5 You do recall how to use L Hospital s rule to compute limits at ifiity right? If ot you should go back ito the Calculus I material do some refreshig. Step We ca see that the limit of the terms existed ad was a fiite umber ad so we kow that the sequece coverges ad its limit is oe. 007 Paul Dawkis 5