Math 133 Series Sequences and series. fa g

1 Math 133 SeriesStewart and any sequence {an} n=1we associate another sequence {sn} n=1, called theseriesof sums of{an}, defined by:sn=a1+a2+ +an=n i= is,s1=a1,s2=a1+a2,s3=a1+a2+a3, and in generalsnis the sum ofthe firstnentries of{an}. The sigma notation ni=1aiis a convenient shorthandfor taking each integer valuei= 1,2,..,n, substituting it into the expressionai,and adding all the resulting quantities (see Pt 2). We can change the indexletters arbitrarily:sN= Nn=1andefines the same Series assn= ni= :Ifan=1n, thens1= 1,s2= 1 +12=32,s3= 1 +12+13=116, general termsn= ni=11i=11+12+13+ +1nhas no elementary formula,which is typical for Series , even whenanis quite simple.

2 Geometrically,snis thearea under the bar graph of{an}above the interval [0,n].We can also take the infinite sum, which is defined as a limit of finite sums: i=1ai= limn n i=1ai= limn limit, converging or diverging, is the total area under the bar graph of{an} n= :The general purpose of a Series is to express a complicated quantityas an infinite sum of simple quantities,s= i=1ai, so that the finite sumssn= ni=1aiare approximations. A familiar example of this is decimal notation:an irrational real number is a complicated quantity with an infinite amount ofdetail in its digits, which are equivalent to the sum of a certain a sequence of digits{dn} n=1withdn {0,1.}

3 ,9}, we have the number:s= =d1101+d2102+d3103+ = n= definition, an infinite decimal is the limit of its finite decimal approximations,the number approached as we add more digits. A trivial example is the repeatingdecimals= , which clearly gets as close as desired to 1 as we take moreNotes by Peter sos= 1. For a more complicated pattern of digits, the Series converges tosome complicated real number (see ).example:We will eventually ( , ) develop powerful methods to writefamiliar numbers and functions as infinite Series . Two outstanding formulas are: = 4(1 12+13 14+ ),sin(x) =x 13!

4 X3+15!x5 17!x7+ .It is formulas like these that allow machines to compute complicated transcenden-tal quantities using only the four arithmetic operations (which are all that caneasily be built into a logic circuit).Wheat kernels on a the following classic puzzle: if weput one kernel of wheat on the first square of a chessboard, then two kernels onthe second square, then four on the third square, and we keep doubling until the64th square, how many kernels on the whole board?We start with the sequencean= number of kernels on thenth square, definedrecursively bya1= 1,an= 2an 1, which leads to the explicit formulaan= 2n is called an exponential sequence orgeometric sequence .

5 The associatedgeometric seriesissn= total number of kernels on the firstnsquares:sn=n i=12i 1= 1 + 21+ 22+ + 2n 2+ 2n , we can find a simple formula forsnas follows:2sn=21+ 22+ + 2n 1+ 2n sn= 1 21 22 2n 1 Adding these, the two sides become:(2 1)sn= 2an a1= 2n 1= sn=2n 12 1= 2n , the answer to our puzzle iss64= 264 1 kernels, which is enough wheatto fill a football we change the problem: we put one ounce of gold on the first square, halfan ounce on the second square, a quarter ounce on the third square, and so onuntil the 64th. What is the total weight of gold on the board?

6 The weight on thenth square is given by another geometric ( exponential)sequencean=(12)n 1, and the total weight on the firstnsquares is the geometricseriessn= ni=1(12)i 1. We use the same trick as before:sn= 1 +12+(12)2+ +(12)n 1 12sn= 12 (12)2 (12)n 1 (12)n(1 12)sn=a1 12an= 1 (12)n= sn=1 (12)n1 12. This terminology is obscure, but very , the total weight iss64=1 (12)641 12. Since(12)64is a tiny, negligeable quantity,this is very close to11 12= 2, meaning the first square has just about the sametotal weight as the other 63 squares. In fact, adding more squares would barelychange the total, since the limit is: i=1(12)i 1= limn sn= limn 1 (12)n1 12=1 01 12= Sequences and general geometric sequence starts with aninitial valuea1=c, and subsequent terms are multiplied by the ratior, so thatan=ran 1; explicitly,an=crn 1.

7 The same trick as above gives a formula forthe corresponding geometric Series . We havesn rsn=c crn, so:sn=n i=1cri 1=c+cr+cr2+ +crn 1=c1 rn1 r.(Notice that the powerrnis one larger than in the last termcrn 1.) This ingeniousformula is known as thesum of a finite geometric Series ; the limit is thesum ofan infinite geometric Series :limn sn= i=1cri 1=c11 r,provided|r|< couse, the infinite Series diverges if|r| 1. These formulas are needed againand again in practical problems, especially those involving finance and sigma geometric Series formulas allow us to eval-uate any Series whose terms involve only exponential functions like 2nor 2 n, butnot power functions likenorn2.

8 For this, we rearrange and manipulate the termsinto the form of the geometric Sequences which we :Evaluate the finite sumn i=12i 13i+1, and the infinite sum i=12i 13i+ , we work this out in dot-dot-dot notation:21 131+1+ +2n 13n+1=(2131+1+ +2n3n+1) (131+1+ +13n+1)=(23221 131 1+ +2322n 13n 1) (132131 1+ +13213n 1)=(232(23)1 1+ +232(23)n 1) (132(13)1 1+ +132(13)n 1)=2321 (23)n1 23 1321 (13)n1 13 Here we factored out232and132so the remaining factor would beri 1for somer, which we can then evaluate using the geometric Series computation can be written more compactly in sigma notation.

9 N i=12i 13i+1=n i=1(2i3i+1 13i+1)=n i=12i3i+1 n i=113i+1=n i=12322i 13i 1 n i=113213i 1=n i=1232(23)i 1 n i=1132(13)i 1=2321 (23)n1 23 1321 (13)n1 get the infinite sum, we just remove the terms (23)nand (13)n, since these go tozero asn .Repeating important application of geometric Series is to writerepeating infinite decimals as fractions. For example, consider:s= = can apply the ratio-multiplication trick to cancel the infinite tail of digits:s= ,1100s= (1 1100)s= 99100s=621000= s= is no coincidence that we can use the same trick as before: in fact, thisinfinite decimal is the sum of a geometric Series .

10 S=6102+2103+6104+2105+6106+2107+ =62103+62105+62107+ = i=162102i+1= i=1621011(102)i= i=162103(1100)i 1=6210311 1100=62100010099= = + =1510+62990= can carry out such reasoning for any infinite decimal which starts with ar-bitrary digits, then becomes repeating. Thus, any infinite decimal represents areal number, but the repeating decimals represent precisely the rational numbers(fractions)! For example, since we know 2 = is anirrational number, not equal to any fraction, its decimal digits will not power can take the ratio in a geometric Series to be avariable, obtaining a function called apower Series :g(x) = n=0xn= 1 +x+x2+x3+ (Traditionally, the index starts atn= 0, so the first term isx0= 1.)