Math 104 – Calculus 10.2 Infinite Series Math 104 - Yu InfiniteInfinite series series •• Given a sequence we try to make sense of the infinite Given a sequence we try to make sense of the infinite sum of its terms. sum of its terms. 1 • Example: a = n 2n 1 s = a = 1 1 2 1 1 s = a + a = + =0.75 2 1 2 2 4 1 1 1 s = a + a + a = + + =0.875 3 1 2 3 2 4 8 1 1 s = a + + a = + + =0.996 8 1 ··· 8 2 ··· 256 s20 =0.99999905 Nicolas Fraiman MathMath 104 - Yu 104 InfiniteInfinite series series Math 104 - YuNicolas Fraiman Math 104 Geometric series GeometricGeometric series series • A geometric series is one in which each term is obtained • A geometric series is one in which each term is obtained from the preceding one by multiplying it by the common ratiofrom rthe. preceding one by multiplying it by the common ratio r. 1 n 1 2 3 1 arn−1 = a + ar + ar2 + ar3 + ar − = a + ar + ar + ar + ··· kX=1 ··· kX=1 2 n 1 • DoesWe have not converges for= somea + ar values+ ar of+ r + ar − n ··· 1 2 3 n rsn =n ar1 + ar + ar + + ar r =1then ar − = a + a + a + a···+ n ···!1 sn rsn = a ar −kX=1 − 1 na(11 rn) r = 1 then s ar= − =− a + a a + a a − n 1 −r − − ··· kX=1 − Nicolas Fraiman MathNicolas 104 Fraiman Math 104 - YuMath 104 GeometricGeometric series series Nicolas Fraiman Math 104 - YuMath 104 Geometric series Geometric series 1 1 1 = 4n 3 n=1 X Math 104 - Yu Nicolas Fraiman Math 104 RepeatingRepeatingRepeang decimals decimals decimals • We cancan useuse geometricgeometric series series to to convert convert repeating repeating decimals toto fractions.fractions. • Example: 11..2323 = = 1 1.2323232323.2323232323...... 2323 2323 2323 11..2323 = = 1 1 + + ++ ++ ++ 100100 1000010000 10000001000000 ······ 11 2323 2323/100/100 11..2323 = = 1 1 + + =1+=1+ 100100nn 11 1/1100/100 nn=1=1 − XX − 2323/100/100 2323 122122 =1+=1+ =1+=1+ = = 9999/100/100 9999 9999 Math 104 - Yu Nicolas Fraiman MathNicolas 104 Fraiman Math 104 Telescoping series • A telescopingTelescopingTelescoping series series is one in which series the middle terms cancel and the sum collapses into just a few terms. • A telescoping series is one in which the middle terms cancel and the sum collapses into just a few terms. • Find the sum of the following series: • Find1 the sum3 of the 3following series: 1. n2 − (n + 1)2 n1=1 ✓ 3 3 ◆ 1. X n2 − (n + 1)2 n1=1 ✓ 3 ◆ 2. X k(k + 3) n1=1 3 2. X k(k + 3) n1=1 1 1 X 3. ln(n + 2) − ln(n + 1) n1=1 1 1 X ✓ ◆ 3. ln(n + 2) − ln(n + 1) n=1 ✓ ◆ X Nicolas Fraiman Math 104 - YuMath 104 Nicolas Fraiman Math 104 Divergence test DivergenceDivergenceDivergence Test test test ! ! ! ! n n 1 n n −1 Proof: an = ak − ak = sn sn 1 L L =0. Proof: an = ak − ak = sn −sn 1− !L −L =0. k=1 − k=1 − − ! − kX=1 k=1X X X Math 104 - Yu NicolasNicolas Fraiman Fraiman MathMath Nicolas104 104 Fraiman Math 104 ExamplesExamples Examples 1 3n2 1. Does converge? 1 n(3nn+2 1) 1. Does n=1 converge? Xn(n + 1) n=1 X 1 2. Does sin( n ) converge? 1 2. Does n=1 converge? Xsin(n) ! n=1 X ! Remark: 1 1 1 The converse is not true 0 but diverges! n n ! n=1 X Nicolas Fraiman MathMath 104 - Yu 104 Nicolas Fraiman Math 104 PropertiesProperties of of convergent convergent series series ! Properes of convergence ! ! ! ! 1 3 1 1 Example: =3 =3 1=3. 2n 2n · n=1 n=1 X X Nicolas Fraiman Math 104 Nicolas Fraiman Math 104 Math 104 - Yu Reindexing series Math 104 - Yu .