Calculus II MATH 1502
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This 0 page Class Notes was uploaded by Chelsea Nolan MD on Monday November 2, 2015. The Class Notes belongs to MATH 1502 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 34 views. For similar materials see /class/233957/math-1502-georgia-institute-of-technology-main-campus in Mathematics (M) at Georgia Institute of Technology - Main Campus.
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Date Created: 11/02/15
Chapter 5 Series and Convergence We know a Taylor Series for a function is a polynomial approximations for that function This week we will see that within a given range of X values the Taylor series converges to the function itself In order to fully understand what that means we must understand the notion of a limit and convergence 51 Limits of functions and L hopital s rule We begin with a brief review of limits You have probably studied limits of functions before lntuitively lim fX L means the closer X gets to a the closer the value fX gets to L Indeed as close as you want to Xgta get to L say within 0001 you can nd an Xquot such that lfX 7 Ll 5 0001 Some limits are easy to nd For example lim X a and lim cf X c lim fX If we happen to know Xgta Xgta x98 that lim fX L1 and lim gX L2 then it is true that Xgta X98 Dig M N L1 L2 ii lim fXgX L1L2 Xgta iii lim fXgX LlLz if L2 0 Xgta You may recall that it is more dif cult to nd lim fXgX when lim fX 0 00 or 700 and Xgta X98 5X5 7 3X lim gX 0 00 or 700 For example consider lim 7 Both the numerator and denomonator Xgta x900 7 5 7 go to 00 but the fraction goes to 57 One way to see this is to graph the function Here39s another method Theorem L Hopital s rule If lim fXgX is ofindeterminate form 00 or ioo i 00 then Xgta hm foogm hm f ltxgtg ltxgt Xgt 5 Here are some examples lim 3X2 47X7 23 lim 6X7 oo x900 x900 lirI QXZ 4X7 20x3 7 8 lim6x 43x2 43 X92 We can now make a general statement about limits of quotients of polynomials Theorem If f X and gX are polynomials then Chapter 5 Series and Convergence 22 oo ifdegfx gt gX lim fXgx 0 if deg X lt gX X900 akbk if deg fX gX where ak and bk are the coef cients of the highest terms in fx adn gX respectively 52 Limits of sequences A sequence is a function whose domain is all positive integers For example 2 4 8 16 is a sequence whose nth term is 2quot We write this as the sequence an 2quot List the rst few terms of the sequence 31 what is the 34th term of this sequence In this class we will mostly be interested in limits of sequences as X a 00 It is not too hard to believe that lim 2 00 and that lim 0 lntuitively we understand that as n gets really big then i gets new new 1 3n really close to 0 Now we try to formalize that idea De nition lim an L if for any 5 gt 0 there exists an integer N so that whenever n gt N it is true that ngtoo laquot 7 Ll lt e We say the sequence an converges to L in this case For example lim E 0 since whenever n gt lSe le inorder to make 31 within 5 001 of0 we ngtoo need to pick n gt 3601 333 53 Series A series is an in nite sum like 1 quot 1 1 2 1 3 1 4 25 1ltEgtltEgt 9 ltEgt 39 10 Taylor Series are examples of series In this section we address the following question If we look at partial sums of the series do the answers we get approach some limit le n 1 139 lim 2 In other words does the sequence of partial sums an 20 converge to some limit We say that a series converges if such a limit exists and is nite and diverges otherwise Here are three examples the rst series converges and the second and third diverge 2 10 112345 Mg l l c 00 2711711711711m 1390 In order for a series to converge it must be true that the terms in it get smaller and smaller More exactly Chapter 5 Series and Convergence Theorem If 20 b1 converges then limi900 b1 0 This condition is needed for a series to converge but is not suf cient to insure convergence So if we have a series 20 b1 and limi900 b1 0 then the series must diverge The second and third examples above are examples of this But it is possible that a series 20 a has limi900 a1 0 and doesn39t converge A surprising example is 1 1 1 l E E Z 54 The Geometric Series The rst series we will talk about is called the geometric series It is of the form 00 Zx lxX2X3X4 10 Notice that the very rst series mentioned at the top of this page is such a series with x Whether this series converges or not will depend on what x is We rst look at the simple case that x It is useful to de ne the partial sums here and study there behavior V H 1 1 5n lt5 1 10 Calculating these we see So l 51 52 53 1T 54 1 13 We see a pattern lndeed 1 quot 2quot1 7 1 1 33 Suite 1121342 For the general geometric series we again look at partial sums Snz2 vlxxzXn 10 In this case some algebra proves useful Consider multiplying the partial sum by l 7 x 17XSH17X1XXZ XH17XHT1 This means that 1 7 Xn1 SH l 7 x So we need to determine the limit 1 7 Xn1 1 7 Notice that it is a limit as 11 goes to in nity The only 11 is in the numerator Whether this converges or not will depend on what x is In the example when x the series converges When x 2 on the otherhand the series will diverge The general rule is that the series will converge as long as lx lt 1 Why So if lx lt 1 lim Sn lim ngtoo ngtoltgt i r 17Xn1 1 x l1mSn l1m ngtOO 00 17X 17X Mg l 2 Check that this corresponds with what we got when x Chapter 5 Series and Convergence 24 Repeating decimals You all know that 3333333333 but here is a proof of this fact 333333373030030003 731313 1 73 1 i 39 739 39 39 39 T 10 100 1000 7 11 10 l l l 3ltm 1gt3lt gt 55 Alternating Series 00 An alternating series is one in which the terms alternate in sign so it will look like Z 7 1 quotbn where bquot will n1 be sequence The following theorem about alternating series will be useful Theorem An alternating series 207libi converges if and only if limi900 b1 0 For example the series 207li converges while the series 207112 diverges Compare this result with the previous theorem This one is quotif and only if quot while the previous theorem was not 56 Tests for Convergence and Absolute Convergence We continue with more ways of determining whether a series converges or not Since we already have a method which determines whether alternating series converge or diverge this week we will concentrate on series of positive terms 561 Method 1 Comparison I If 3 a1 and L73 b1 are series of ositive terms and a1 5 b1 and the series 3 b1 conver es then so 10 10 P 10 g does the series 20 a II If 3 a1 and g3 b1 are series of ositive terms and a1 3 b1 and the series 3 b1 diver es then 10 10 P 10 g so does the series 20 a1 EXAMPLE Show that the series 210 141 converges We compare this series with the series 20 which is a geometric series that converges to 2 We compare the ith terms 1 i 1 lt 1 1 11 123111 12222 2139 562 Method 2 Integral Test If each a fi for some continuous decreasing function fX then 00 00 Zai converges ifand only if fXdX converges 11 1 EXAMPLE Show that the series diverges The function fX i is continuous and decreasing on the interval 1 00 and a1 0 1 l dX lim dX limlnm71nloo 1 X 11900 1 X 11900 Chapter 5 Series and Convergence 25 1 EXAMPLE The series converges The function fx F interval 1 00 and a1 0 1 r m l r l 1 dx 11m dx 11m 17 71 1 xP mace 1 xP mace 1 7 p m P This limit is nite ifp gt 1 and in nite otherwise is continuous and decreasing on the THEOREM The series converges ifp gt 1 and diverges ifp 5 1 Absolute Convergence Sometimes a series is neither all positive term nor nicely alternating in sign An example is C gzs l some of whose terms are negative some positive but not every other one In this case we sometimes talk about the absolute convergence of a series The series a1 is said to converge absolutely if the series of the absolute values of the terms 00 Dan la1l lazl 11 converges It is important because of the following result THEOREM If a series converges absolutely then it converges EXAMPLE Looking at our example must converge since converges by comparison with 563 Method 3 Ratio Test This test is a generalization of the comparison test above This tests for absolute convergence THEOREM 00 Z laill a1 converges1fandonly1f 11m lt 1 11 1900 lail EXAMPLE Recall the Geometric series is 1 X x2 c1 The ratio test looks at the ratio of the terms aHl x11 and a1 xi 3 Xi1 lim 1 1 1im 1 1X1 1900 lail 1900 1X11 This limit exists and is lt 1 exactly when 1x1 lt 1 Thus the geometric series converges when 1x1 lt 1 which agrees with what we had determined before 57 Intervals of Convergence This brings us to another de nition Series that contain a variable x say may converge for only some values of X The values for which the series does converge are collectively called the interval of convergence for that series For example the geometric series has interval of convergence 71 lt x lt 1 Returning to the Taylor series we often want to know for what values of x does the Taylor Series of the function converge to the function itself This will always turn out to be an interval around the point a where we centered the Taylor Series EXAMPLES Chapter 5 Series and Convergence 26 i 221 nquotX The ratio test give us that lan1l ln1n1Xquot1l ln1n1Xl lnn1Xl 11m 11m 11m gt 11m gt 11m lnXl H900 lanl H900 ln x l H900 ln l H900 ln l H900 which is 00 unless X 0 So the series converges only if X 0 ii 221 The ratio test shows 2 nlxnl 2 an1 7 n 7 i X 7 Lee W HLOQ 7 HILIE02H1 0lt139 H So this series converges for all X That is the interval of convergence is foo lt X lt 00 58 Problems for Chapter 5 Exercise 51 Find the limit in each case 2X3 7 3X2 4X2 3X 3X 3 x1330 7X 7 13 b x1530 16X3 7 1000 C x1330 2X lnX 2quot sin X 01 1213 T lt6 1213 Xssmaz lt0 21 T Exercise 52 Each of the following sequences converge to 0 In each case nd N as a function of e as the formal de nition of convergence requires a an 17 b an c an Zl n hint use logs Exercise 53 Decide if each of these geometric series converges and if so determine what it converges to a 100100010000 b 20gy cZ01 Exercise 54 Find the rational numbers represented by each of the repeating decimals below a 040404040404 b 123123123123123 Exercise 55 Find a formula for the sum of each of the following series by performing suitable operations on the geometric series a 17X3X67X9 b X2X6X10 C172X3X274X3 X2X23X34X4 exx724 1 AAAA D Exercise 56 Which of the following alternating sums converge and which diverge 00 V 14 00 V 1 00 v i a fl 1 b fl 1 c fl ltgtglt gt121 HQ gt TH 0 gt11 00 i i 00 i 1 d gel m e gel E Chapter 5 Series and Convergence 27 Exercise 57 Use the ratio test to nd the interval of convergence for each of the following power series o2 Z Z 11 39 00 Zm ii min 1391 1391 11 g ilt1 20quot h f i ink 11 n1 n1 Exercise 58 Find the interval of convergence for the Taylor series of the following functions a sinx b equot c cosx Exercise 59 Use the Comparison Test to determine whether each of these series converges or diverges 139 axinnf mn oxide Exercise 510 Use the integral test to determine if each of the following series converges or diverges a b C gaz 01 02 3 of 1 of 1 oil d e f 1 n 1 11 iln i quot1 n Exercise 511 Use the ratio test to nd the interval of convergence for each of the following power series 00 X 00 X 00 X 2 oz ozj n1 n n1 n1 1739 00 00 Hz 00 3 2w o2quot o2 n1 n1 n1 00 00 9quot 00 g lt1 20quot h Zn Xquot i anx n1 Exercise 512 Find the interval of convergence for the Taylor series of the following functions a sinx b equot c cosx
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