. Ifiite Series Oce we have a sequece of umbers, the ext thig to do is to sum them up. Give a sequece a be a sequece: ca we give a sesible meaig to the followig expressio? a = a a a a While summig ifiitely may terms may seem strage, what is t strage is to sum fiitely may terms. We therefore make a defiitio. Defiitio. Let a be a sequece. The th partial sum of the sequece is the value s = a i = a a a i= Example Suppose that a = so that a =,, 8, 6,... We compute the first few partial sums: s = a = s = a a = s = a a a = 7 8 s = a a a a = 5 6 What we are costructig is a ew sequece s, the sequece of partial sums. It certaily looks like we have a formula for the th term of this sequece s = Moreover, the limit of the sequece is lim s =. It seems reasoable to defie this to be the ifiite sum of the origial sequece, ad we write = I geeral, we have the followig: Defiitio. Let a be a sequece. The ifiite series a is the limit of the sequece of partial sums, if it exists. That is a = lim s We say that the series coverges or diverges if ad oly if the above limit does.
Zeroth terms ad otatio All of our previous discussio could have bee performed with a sequece startig with a differetly idexed term, i.e. a =m = a m, a m, a m,.... The most commo choices are m = 0 or. The term a 0 is ofte referred to as the zeroth term of the sequece. If we kow what the iitial term of a sequece is, or if a result does ot deped o the iitial term, the it is commo to omit the limits etirely ad simply deote the series by a. Series Laws Series behave exactly like fiite sums. Therefore, if the series a ad b coverge, ad if c is a costat, we have. ca = c a compare the fiite sum ca ca = ca a. a b = a b compare a a b b = a b a b These expressios are aki to the limit laws. By cotrast with limits, series do ot obey laws regardig products ad divisio. Thus a b = a b ad a b = a b Agai, if you imagie what these would mea for fiite sums, there is o reaso to expect equality! Geometric Series Perhaps the most importat family of ifiite series are those obtaied by summig a geometric sequece: that is a sequece of the form a = ar where a ad r are costats. The motivatig example above is such a sequece, with r =. We ca deal with these i geeral. There is somethig of a covetio regardig the first term of a geometric series, so it is worth makig a ew defiitio. Defiitio. A geometric series is a ifiite series of the form ar for some costats a = 0 ad r. I particular, the sequece a starts with the zeroth term a 0 = ar 0 = a, where we follow the covetio that r 0 =. Theorem. The geometric series with th term a = r coverges if ad oly if < r <, i which case r = r Proof. Suppose first that r =. I this case the th partial sum is simply s = } {{ } = times Clearly the sequece s diverges to ±, depedig o the sig of a, whece a diverges. Now suppose that r =. Cosider the th partial sum. s = r r r r Clearly a a b b = a b a b ad a a b b = a b a b
Multiply this by r. rs = r r r r r Now subtract oe lie from the other, oticig how almost all the terms come i cacellig pairs: rs = r We have therefore obtaied a th term formula for the sequece of partial sums s = r r As we saw i the previous sectio, lim r coverges if ad oly if < r. Sice r = i this case, we also see that this limit is zero, which gives the result. Examples. = = =.. If the summatio does ot start with a zeroth term, the it is a good idea to re-idex so that it does. I what follows, we let m =. = m = / = 7 m=0 If the above feels too fast, try writig out the first few terms of the series: e.g. = = [ ] = m m=0 If you thik about the iitial term you should t go wrog!. Sometimes a little more work with expoetial laws is required i order to view a series as geometric. = 8 = 8 = 9 8 6 9 6 9 = 9 9 m 6 8 6 m=0 6 = 8 8 6 7 = 896. Whe a geometric series diverges it is very easy to spot. For istace =5 = 9 =5 8 diverges to ifiity sice 9 8 >.
Covertig a repeatig decimal ito a fractio repeatig decimals. For example, Geometric series are also useful for uderstadig.5555... = 5 00 5 00!5 = 5 00 = 5 = 0 00 00 = 5 00 00 Ideed it is a theorem that every decimal which evetually has a repeatig patter must be a ratioal umber. Try to covice yourself usig geometric series that.57777... = 0657 99500 I particular this shows that the decimal represetatio of a irratioal umber such as or π will ever have a repeatig block of digits! Telescopig Series Beyod geometric series, there are very few series that we ca compute exactly. Oe such family are kow as telescopig series, ad the idea for how to deal with them is aalogous to the partial fractios method for itegratio. Here is a example: to compute the value of the ifiite series we first cosider the th partial sum: s = = i= It follows that ii = i i i= = = lim s = For a more complicated example, cosider = partial fractios decompositio = cacel terms i pairs
with th partial sum satisfyig s = from which = = lim Showig Divergece 5 6 = It is ofte easier to prove that a series diverges tha to prove covergece, as the followig result shows. Theorem th-term/divergece test. If a does ot coverge to zero, the a does ot coverge. Proof. We prove the cotrapositive. Assume that a coverges to s. The s = lim s where s is the sequece of partial sums. But the Example lim a = lim s s = s s = 0 lim diverges sice = e = 0 No-example The th-term test oly works i oe directio! As we shall see i the ext sectio, the harmoic series diverges, eve though the sequece coverges to zero. Suggested problems. Fid the sum of each of the followig series: a b If a coverges the a coverges to zero. This statemet is logically equivalet to that i the Theorem. 5
. Express the decimal.555 as a fractio.. Suppose you borrow $0,000 for a ew car at a mothly iterest rate of 0.5%. Suppose you make paymets of $600 per moth, paid at the ed of each moth. a Let a be the amout you owe at the start of the th moth. Show that a =.0075a 600. b Let b = a 80000. Fid a recurrece relatio for b ad solve it. c After how may moths will the loa balace be zero? I.e. at the ed of the first moth you owe a extra $00 iterest. 6