Lesson 12. Sequences and Series


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1 Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or simply { a }. Because the th term of a sequece is sufficiet to express the patter of a sequece, it is usually called the geeral term of the sequece For istace, if cosiderig a sequece,,,,,..., the patter of this sequece is determied by the th term of the sequece. I this case, you will aturally try to ivestigate the patter of the umerators, deomiators ad the alteratig sigs i order to fid the expressio of the th term. I this case for =,,,, 5,..., you ca fid The patter of the umerators is {,, 5, 6, 7, }, 5 The patter of the deomiators is { 5, 5, 5, 5, 5, }, 0 5 The patter of the sigs is {( ), ( ), ( ), ( ), ( ), ( ),...( )...}, So the th term of the sequece is ( ) + 5, ad this sequece is represeted by + ( ) 5. A ifiite series is the sum of a ifiite sequece ad writte as = th partial sum of a ifiite sequece is give by a a = a + a The a = a + a + +. For istace, for a ifiite i a sequece,,,,,..., the sum of the sequece is the wellkow geometric series i 8 6 with ratio of /. Whe we studyig calculus, our major cocer is whether a series is coverget or diverget. As you kow, if the limit of the th partial sum s = a + a + a + + a exists, the series is said to be coverget; Otherwise it is diverget.
2 I this lesso we will lear how to use Mathematica to costruct ifiite sequeces, calculate the limit of a sequece, fid the th partial sum of a sequece ad ivestigate the covergece of a ifiite series. The followig four examples are chose to help you practice ad uderstad the computer programmig techiques. Example List the first 7 terms of a geometric sequece with ratio of ½. Fid the sum of the first terms of a geometric sequece with ratio of ½. Determie whether the geometric series is coverget or ot. Solutio To list the first 7 terms of the geometric sequece, we use the commad Table[ ] to list the umbers i order ad the otatio {i,, 7} to idicate the idex i takig the value from to 7.,,,,,,,,, To fid the sum of the first terms of the geometric sequece, we use the commad Sum[ ] ad iclude the otatio { i,, } that idicates the idex i takig its value from to.,,, The outcome geerated by Mathematica is right but looks a little weird. If simplifyig a little bit, the outcome will be like ( + ) =, which is the same as what you leared i calculus class. To determie whether the geometric series sequece from i = to i =. i coverges, we ca fid the sum of all terms i the,,, Sice the limit of the th partial sum is equal to, we ca say that the geometric series is coverget.
3 Example List the first 7 terms of a geometric sequece with ratio of /. Fid the sum of the first terms of a geometric sequece with ratio of /. Determie whether the geometric series is coverget. Solutio This example is similar to the previous oe. You are asked to ivestigate the same iformatio of a geometric sequece with the ratio of /. So we follow the same steps as the previous example. The sum of the first terms is,,,,,,,,,,,, The sum of all the terms of the sequece is,,, Sum::div: Sum does ot coverge. á Sice the limit of the th partial sum is diverget, we ca say that the geometric series is diverget. The covergece or divergece of a geometric series with a ratio of r depeds o the value of the ratio r. We ca use the Mathematica fuctio,,, to fid out the th partial sum of a geometric series. The th partial sum is. Use what we leared from Calculus II, (a) Whe r <, the limit of (b) Whe r, the limit of (as (as r ) is equal to. The series is coverget. r ) is equal to or. The series diverges. Next we will preset the th partial sum of a very special type series Iteger Power Series. The followig four series are chose here for your ivestigatio.
4 Example Fid the sum of the first term of the iteger power sequece. a) i = b) i c) i d) i Solutio = + = + = We simply use the commad Sum[ ] to fid the th partial sum of ay power series i secods. As you kow, these are importat formulas we have leared i calculus. You ca take advatage of Mathematica programmig to fid them wheever you eed them. a) b) c),,,,,,,,, d),,,
5 I the last part of this lesso, a more complicated sequece is chose for you to practice Mathematica programmig oe more time o the commads: Table[ ], Sum[ ], ad Limit[ ]. Example Cosider the sequece + ( ) 5 =. a) List the first 7 terms of the sequece. b) Determie whether the sequece is coverget. c) Fid the sum of the first 7 terms of the sequece. d) Fid the sum of the first terms of the sequece. e) Fid the sum of the etire sequece (from to ). Solutio. a) We list the first 7 terms of the sequece usig the commad Table[ ].,,,,,,,,, Note {,, 7} idicates the idex startig from ad edig at 7. You ca always modify the startig ad edig values of if you eed to. If you wat to sum up all terms from = 0 to = 8, you ca just chage the idex rage to, 0, 8 so that you get,,,,,,,,,,, The it gives the first 9 terms of the sequece with the idex startig from 0 ad edig at 8. 5
6 b) To determie whether the sequece is coverget or diverget, we calculate the limit of the th term by lettig., c) The sum of the first 7 terms of the sequece is calculated by,,, As we discussed before, if you eed the umerical value istead of the fractioal value, we ca always use N[ ] to get d) The sum of the first terms of the sequece is 0.7,,, The Mathematica outcome may ot be the same as what you calculated by had, but it is correct with o doubt. It just seems a little weird compared to what you ormally did i calculus. e) The sum of the etire sequece is calculated by ruig the terms from = to =,,, 6
7 I this case we ca say this series is coverget because the ifiite sum is fiite. Retur to List of Lessos 7
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