5 Sequences and Series

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1 Bria E. Veitch 5 Sequeces ad Series 5. Sequeces A sequece is a list of umbers i a defiite order. a is the first term a 2 is the secod term a is the -th term The sequece {a, a 2, a 3,..., a,..., } is a sequece ad we deote it by Examples of Sequeces {a } or {a } = ) 2) 3) { 2 } {( 2 + 2) } ( + 3) { ( )} 2π cos } 2 +,... } { a = , 4 5, 9 0, 6 7,..., 2 ( a = { 4 ( + 3) 2) 2, 5 4, 6 8, 7 6, 8 32,... ( ) { 2π a = cos,, 2, 0, cos(2π/5), 2, cos(2π/7), 2 2,... } Some sequeces are t defied so clearly. Here s a example of a sequece that s defied recursively. Recursively meas the ext term i the sequece is determied by previous terms. The ext sequece is probably oe of the more famousj sequeces. Example: Fiboacci Sequece {f } defied recursively. f = 2. f 2 = 3. f 3 = 2 24

2 5. Sequeces Bria E. Veitch 4. f 4 = 3 5. f 5 = 5 6. f 6 = 8 7. f = f + f 2 The ext term i the Fiboacci Sequece is the sum of the previous two terms. Defiitio 5. (Limit of a Sequece). A sequece {a } has the it L ad we write a = L or simpliy a L as if we ca make the terms of a N as close to L as we make sufficietly large. If a exists, we say it coverges. Otherwise, we say it diverges. Defiitio 5.2 (The Formal Defiitio of Covergece). We will ot really use this defiitio. It is, however, good to kow. A sequece {a } has a it L if for every ɛ > 0, there is a correspodig iteger N such that if > N, the a L < ɛ This basically says, If you believe the it of this sequeece is L, the fid me the Nth term such that every term after a N is withi ɛ of L. Example 5.. Let a =. I claim the it is 0, i.e., a 0 as. Let ɛ =.. My job is to fid the term where a is fially withi 0. of L = 0. a = N = 0. 25

3 5. Sequeces Bria E. Veitch 0. = N N = 0 So after N = 0, every term of a is withi 0. of L = 0. {, 2, 3,..., 9, 0,, 2, 3, 4,..., } Do you see how after the 0-th term, a is withi 0. of L = 0? Defiitio 5.3 (Divergece). =, meas that for every positive umber M, there is a iteger N such that if > N, the a > M. This defiitio basically says, If you claim the it is, the I ll give you a very large umber (like 00,000,000). You eed to tell me the term i a where a > 00, 000, 000. By doig this for ay extremely large umber, you essetiall prove a = Let s talk Limit Laws If a ad b are coverget sequeces ad c is some costat:. a ± b = a ± b 2. c a = c a 3. a b = a b a 4. = a b b 5. (a ) p = ( a ) p if 0 Before we do some examples, we have two very useful theorems. 26

4 5. Sequeces Bria E. Veitch 5.. Sequece Theorems Theorem 5. (The Squeeze Theorem). If a b c for N ad a = c = L, the Aother (ot amed) theorem b = L If a = 0, the a = 0 Ok, ok.. ow some Examples Example 5.2. Write out the first five terms of a = { 2 2, 4 5, 6 0, 8 7, 0 } 26,... It appears the sequece coverges to 0, a 0. Example 5.3. Write out the first five terms of a = ( ) ( + )! Recall that! = ( ) ( 2) ( 3) { 2, 6, 25, 20, } 720,... This is called a alteratig sequece sice the sigs alterate betwee ad. It also appears the sequece coverges to 0. 27

5 5. Sequeces Bria E. Veitch Example 5.4. Write out the first five terms for the sequece that s defied as a = 6, a + = a Note that = does t mea a, = 2 does t mea a 2. a = a 2 = a + = a = = a 3 = a 2+ = a 2 2 = 2 a 4 = a 3+ = a 3 3 = /2 3 = 6 a 5 = a 4+ = a 4 4 = 24 Example 5.5. Fid + Most of these ca be doe usig methods we already kow. + = = + 0 = + Just to verify this, let s look at the first few terms ad see if it appears the sequece approaches. l Example 5.6. Fid { 2, 2 3, 3 4, 4 5, 5 6, 7 } ,...,,..., ,... 28

6 5. Sequeces Bria E. Veitch Ooooh, I thik we have a rule that helps with this. What was it called...l Hospital s Rule! l LH / = 0 Example 5.7. Is a = ( 2) coverget or diverget? {, 2, 4, 8, 6, 32, 64, 28, 256,...} It appears a is divergig. The umbers keep gettig larger ad larger. They are also alteratig. Some of the terms appear to approach while the others. This really just adds more to why this sequece diverges. Example 5.8. Does a = ( ) coverge? Yes. If you look at the sequece {, 2, 3, 4, 5, 6,..., 00, } 0,... It appears to coverge to 0. So how ca we prove that? Let s use our u-amed theorem from a couple of pages ago. a = ( ) = We kow 0 as. Therefore, sice a = 0, a must coverge to 0. 29

7 5. Sequeces Bria E. Veitch Example 5.9. Let a = cos(π/) cos(π/) = cos(0) = Example 5.0. Where is a = r, r is a costat, coverget? Through a little work, which I ll show you i class, we have r = 0, if < r <, if r = This is called a GEOMETRIC SEQUENCE. Example 5.. Some quick examples ( ) 2. = 0 sice < < ( ) = diverges because > 3. = sice r = ( ) = 0 5 coverges to 5 8 Example 5.2. Does a = 2 4 Let s check its it. 7 coverge? 220

8 5. Sequeces Bria E. Veitch 2 4 LH 2 4 l l 7 LH 2 4 (l 4) 2 7 (l 7) 2 LH 4 (l 4) 3 7 (l 7) 2 ) 3 = ( l 4 l 7 ( ) 4 7 = 0 sice < 4 7 < So yes, a = does coverge. Example 5.3. Does a = 5+4 coverge? 3 2 We eed to rewrite this so it s of the form r = = ( 5 3 = sice 5 3 > ) Example 5.4. Does a = cos2 () 2 coverge? We kow Divide all sides by 2 cos 2 (). Sice 2 = 2 cos2 (x) 2 2 = 0, the by the Squeeze Theorem, 2 cos 2 () 2 = 0 22

9 5. Sequeces Bria E. Veitch Oe of the biggest skills you ca develop durig the sequece ad series sectio is to be able to look at a sequece ad make a educated (ad hopefully) correct guess o whether the sequece coverges. Here s a list of terms you ll come across ad how they rak amog each other whe is very large. c < l < ay polyomial < expoetial fuctios (base > ) <! < A example would be Example 5.5. Does a =! coverge? 5 < 5 l < 3 <! < Based o what I wrote above, this should coverge to 0. But let s go ahead ad prove it. Let s take a look at some of the terms. a = a 2 = a 3 = Followig the patter, we ca write a = ( )( 2)( 3) Each term o the umerator pairs with a o the deomiator. a =

10 5. Sequeces Bria E. Veitch Rewrite it as a = a = ( ) Sice every product i the paretheses is, if you get rid of them, we have By the Squeeze Theorem, sice 0 < a = 0, we have! = 0 We re almost doe. Let s just go over a couple more defiitios before we move o. Defiitio 5.4. A sequece is icreasig if a < a + for. A sequece is decreasig if a > a + for. A sequece is mootoic if it is always icreasig or always decreasig. Example 5.6. a = If you write out the first few terms, you ll see a is decreasig to 0. So a a decreasig mootoic sequece. a = The first few terms are { 3 6, 3 7, 3 8, 3 9, 3 } 0,..., 223

11 5. Sequeces Bria E. Veitch {0, 2, 23, 34, 45, 56, 67,... } So a coverges to. Sice it s always icreasig, it s mootoic. a = ( ) The first few terms are {, 2, 3, 4, 5, 6, } 7,... We ca see a coverges to 0. But it s ot always icreasig ad ot always decreasig. This is a example of a o-mootoic sequece. Defiitio 5.5. A sequece is bouded above if there is a umber M such that a M for. A sequece is bouded below if there is a umber m such that a m for. A sequece is called bouded if it is bouded from above ad below. Example a = is bouded below by 0 ad above by. 2. a = ( ) is bouded below by -, ad above. 3. a = is boude below by, but it is ot bouded above sice = 224

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