ftz]}]z .tt#t*qtmjfi aiii } { n } or [ n ] I anianforn 1+2=2+-5 an = n 11.1 Sequences A sequence is a list of numbers in a certain order:

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1 Sequences A sequence is a list of numbers in a certain order: a 1 a 2 a 3 a 4 a n1 a n (In this book all the sequences will be infinite.) The number a 1 is called the first term of the sequence. Other ways to write sequences: {a n }! " $ = 1 Ex: Find a formula for the general term a n of these sequences: 1) { } an = n aiii } { n } or [ n ] I 2) n ' *+ ( *. / ( ( (.( '( I in.tt#t*qtmjfi } 3) { } (Fibonacci) ftz]}]z 1+2=2+5 a. i I anianforn > are 2

2 A series is called increasing if a n < a n+1 for all n > 1 A series is called decreasing if a n > a n+1 for all n > 1 A sequence that is either increasing or decreasing is called monotonic. Ex: Show that the sequence a n = " " 1 2 is decreasing. Let an > An + n# ' ni n # > I +z YNIEII III! ' n' f n 't n > I for h > I read this way

3 Convergence of Sequences A sequence {a n } has the limit L and we write lim! " = L or a n L as n " if we can make the terms a n as close to L as we like by taking n sufficiently large. If lim "! " exists we say the sequence converges (or is convergent.) Otherwise we say the sequence diverges (or is divergent.) (There s also a delta/epsilon definition which we ll skip. You re welcome.) lim "! " = means that for every positive number M there is an integer N such that if n > N then a n > M Page 693 lists the Limit Laws for Sequences which cover addition subtraction multiplication by a constant multiplication of two sequences division of sequences and exponents. (They work just like the usual Limit Laws.)

4 The Squeeze Theorem If a n < b n < c n for n > n 0 and lim! " = lim 9 " = : " " then lim " ; " = L If lim "! " = 0 then lim "! " = 0 If lim "! " = : and the function f is continuous at L then lim =(! ") = f(l) "

5 " Ex Find lim " "2 We re going to use the technique of dividing all terms by the highest power of n: = hiss = 1 Ex: Evaluate lim (*)@ " " an = ELI Ian 1 = ht if it exists so Liao lat Ex: Find lim " sin nlisma Lisa sin (F) D " = = tight 10 o sinks F) = sin [ 0 ] = 0

6 ... Ex: When will {r n } be convergent? Let s look at: r = 1 r = 2 r = 1: 11 r = 0.5 : : Yes... b NO... No Yes The sequence {r n } is convergent if 1 < r < 1 and divergent for all other values of r. lim " E" = 0 F= 1 < E < 1 1 F= E = 1 A sequence is bounded above if there is a number M such that a n < M for all n > 1 A sequence is bounded below if there is a number m such that a n > m for all n > 1 If the sequence is bounded above and below it s a bounded sequence. Every bounded monotonic sequence is convergent. See page 698 for a nottoohard proof. (If time page 699 for induction proof.)

7 Series A series is the sum of a sequence and an infinite series is the sum of an infinite sequence. Any number can be written as an infinite sum. Ex: Write pi = as an infinite sum. pi = 3 + I + + I 1 + at + # +. The rest of Chapter 11 deals with finding the sums of infinite sequences and then gives a few applications of them. T "J $ = Sigma Notice that lim " $ = so the sum will get too big to converge to a finite value. "J = ' ( + " (This is the Harmonic Series it is famously divergent) Notice that lim = 0 but the partial sums " " s 1 = a 1 = 1 s 2 = a 1 + a 2 = 1 + ½ = 3/2 s 3 = a 1 + a 2 + a 3 = 1 + ½ + 1/3 = 11/6 don t converge to a finite number Sequence : S I ± 52 Ss.

8 . "J = 1 a K. (This is a Geometric Series) In this case lim = 0 and the partial sums converge to 1: s 1 = a 1 = ½ s 2 = a 1 + a 2 = ½ + ¼ = ¾ s 3 = a 1 + a 2 + a 3 = ½ + ¼ + 1/8 = /8 s 4 = a 1 + a 2 + a 3 + a 4 = ½ + ¼ + 1/8 + 1/16 = 15/16 In general the Geometric Series given by NJO LM N*O = a + ar + ar 2 + ar 3 + is convergent if r < 1 Its sum is given by "J!E "* = P *Q (as long as r < 1) Otherwise the series is divergent. Formula Hell Ex: Find the sum of this series: 5 I + I +I '= = + ' S / T a 5 5 G) +5 ( ) (E) * E. arm E =Fe= =?

9 Theorem: If the series "J! " converges then lim "! " = 0 Ex: Show that the series " 1 "J diverges. (" 1 2+ Telescoping Sums Ex: Show that the series and find its sum. "J is convergent "("2) The trick here is to use partial fraction decomposition: so then we get s n = " TJ = T(T2) = P(P2) P P2 " TJ = T T2 1 + ' + ' + + " "2 so s n = which means that lim " U " = lim " So the series converges and "J "("2)

10 11.3 The Integral Test and Estimates of Sums Ex: Determine whether "J converges or diverges " 1 Unfortunately there s no simple formula for the partial sums s n so it s hard to determine whether the sums converge or diverge (although the book gives a chart which suggests that the partial sums approach 1.64.) Looking at the series we notice that the terms look a lot like a Riemann Sum: ' 1 + and then we realize that as long as we remove the first rectangle (and effectively start at x = 1) the area under the curve is very similar to the sum of these rectangles and we get the following: The Integral Test Suppose f is a continuous positive decreasing function on [1 ] and let a n = f(n). Then the series "J! " is convergent if and only if the improper integral = W XW is convergent. In other words: a) If = W XW b) If = W XW is convergent then "J! " is convergent. is divergent then "J! " is divergent.

11 Ex: Use The Integral Test to determine whether this series is convergent: " "J " 1 2 (Question: Is it decreasing? Hint: Look at the derivative) The pseries if p < 1. O NJO is convergent if p>1 and divergent N Y Ex: Is this series convergent: 1 + K + /

12 11.4 The Comparison Test(s) Sometimes we have a series that we re pretty sure converges even though it s hard to show. For example O NJO = Z N O + O + O + O + Z [ \ O] converges because it s a geometric series with a = ½ and r = ½. But what about the series O NJO = Z N 2O O + O + Ò + O +? ^ _ Oa Even though it s not a geometric series each term is smaller than the terms in the geometric series above it so it stands to reason that the sum will be smaller too and therefore converge. This is formalized in the Comparison Test. The Comparison Test Suppose that! " and ; " are series with positive terms. Then i) If ; " is convergent and a n < b n for all n then! " is also convergent and ii) If ; " is divergent and a n > b n for all n then! " is also divergent. Ex: Test the series divergence. bc d dj for convergence or d bc d (Hint: Compare dj to the Harmonic Series d which is divergent.) dj d

13 11.5 Alternating Series

14 11.6 Absolute Convergence and the Ratio and Root Tests

15 11. Strategy for Testing Series

16 11.8 Power Series A Power Series is a series of the form "JI 9 " W " = 9 I + 9 W + 9 W + 9 ' W ' +

17 11.9 Representations of Functions as Power Series

18 11.10 Taylor and MacLaurin Series

19 11.11 Applications of Taylor Polynomials