Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to this one. Notice the consecutive powers of r, nd wht hppens if we multiply by r. So S n r S n = n which upon division by (Note: 0 ) implies tht Note ll we need is r 1 nd multiply the previous identity by ny rel number '' nd we obtin: Definition 0.1: (Geometric sum) Let, r be rel numbers. Let n N. A SUM of the form is clled FINITE GEOMETRIC SUM. If r 1 then the bove sum is equl to Note if r is 1 then the sum is simply n. Exmple Motivting infinite series A rubber bll is dropped from height of 1m. On ech bounce, it trvels.7 of its previous height. Wht is the totl distnce the bll trvels DOWNWARD. It would be nice to sy the bll trvells nd not worry bout the fct tht we re trying to dd togethe infinite number of things, which is of course impossible to do, even for computer. So to nswer this we will think bout the totl distnce (DOWN) the bll hs trvelled when it strikes the ground for the first time, nd then the totl distnce the bll hs trvelled when it strikes the ground for the second time, nd so on. Clerly then we re considering SEQUENCE of distnces. In fct the sequence of distnces is S n = 1 + r + r 2 + r 3 + + 1 S n = n + r + r 2 + r 3 +... + 1 n. 1 +.7 +.7 2 +.7 3 +.7 4 +...
This is n infinite sequence, where ech term of the sequence is the term before it with nother little bit dded on. Does this sequence converge? A generl technique to nswer this is to find formul tht gives the terms of the sequence, nd tke the limit of the formul s n goes to infinity. Such formul, since it describes the 'generl' oth term of the sequence, will probbly involve 'n'. The nth term of the SEQUENCE bove is It isn't esy to tke the limit (s n of this formul. But our previous work gives us n expression equivlent to this one, nmely: Our tool from before cn be esily pplied here, since this formul is mde up of little prts tht ll converge s n. Hence the limit is simply In prticulr, the sequence converges. We could hve n interesting philisophicl discussion bout how close this MODEL of wht is hppening to the bll is to relity. I will only point out tht just becuse there re n infinite number of 'bounces' it doesn't men the bll never stops. Perhps the time between bounces is lso decresing very quickly. See Zeno's prdox. Certinly it is not unresonble to sy the totl distnce the bll trvels downwrd is the limit of this sequence. Infinite SEQUENCES like the bove occur more thn you might expect in modelling the rel world. Our work so fr cn be generlized s follows: Theorem 0.2: Let, r be rel numbers, with r ( 1, 1). Then the infinite SEQUENCE hs n nth term equivlent to This sequence converges to (1, 1 +.7, 1 +.7 +.7 2, 1 +.7 +.7 2 +.7 3,... ) 1 +.7 +.7 2 +.7 3 +... +.7 n 1 1 0 1.7 1 (.7) n 1.7 1 10 = =..3 3 (, + r, + r + r 2, + r + r 2 + r 3,... ) n.. 1
Intuitively, the bove sequence rose in trying to dd togethe infinite number of things. This sitution occurs often enough to be given nme. Definition 0.3: (Infinite Geometric Series) The nottion (Note -- no comms. Trying to dd togethe infinite number of things.) which we lso write s is clled n INFINITE GEOMETRIC SERIES. Although we use sigm nottion, this nottion simply refers to the SEQUENCE When we sy 'the infinite geometric SERIES converges', we simply men tht the SEQUENCE ssocited with the series (the bove sequence) converges. Combining our results we find + r + r 2 + r 3 + r 4 +... n=0 (, + r, + r + r 2, + r + r 2 + r 3,... ) Theorem 0.4: Let, r be ny rel numbers with r ( 1, 1). Then the infinite geometric series n=0 converges to. This is true becuse the SEQUENCE ssocited with the nottion, nd tht is ll we re relly sying. 1 r 1 r n=0 converges to Exmple Exmple of using the bove formul with 'cookie' series, Exmple Exmple with 1/2 + 1/4 + 1/8+... 6 + 3 + 3/2 + 3/ 2 2 +... (Is this series Geometric SERIES? Is the next term just the previous term multiplied by some 'r'?) Wht is 'r' nd wht is ''? Perhps things re clerer if we write series (equivlently!) s
6 + 6/2 + 6/ 2 2 + 6/ 2 3 +... Infinite Series There re other wys we might try to dd togethe infinite number of things. For exmple is n interesting cse s we will see. It might be best to think of everything in terms of ddition, nmely Definition 0.5: (Infinite Series) Given ny rel numbers lso written we cn try to 'dd them ll, the nottion is clled n INFINITE SERIES. This nottion is simply nottion for the following SEQUENCE, clled the "SEQUENCE OF PARTIAL SUMS OF THE SERIES", nmely: When the sequence of prtil sums converges to rel number L, we lso sy the series converges to L, written, 1 + +... 1 + ( ) + + ( ) + +... More generlly, given ny (fixed) rel numbers 1, 2, 3,... up'. More formlly: i n=1 + + + +... 1 2 3 4,,,... 1 2 3 (, +, + +, + + +,... ). 1 1 2 1 2 3 1 2 3 4 i n=1 It is importnt to understnd tht when we sy n infinite SERIES converges wht we MEAN is tht the ssocited SEQUENCE of prtil sums converges. (And similrly for Divergence -- not converging.) One wy to think of this is tht you cn't dd up n infinite number of rel numbers, so first just try dding up the first number, then try dding up the first two numbers, then the first three, nd so on, nd consider the sequence of prtil dditions, or prtil sums, you hve done., = L.
Fct: the infinite series converges to π 4 1 + ( ) + + ( ) + +.... We will gin some insight into this lter in the course.