Quiz No.. Defie: a ifiite sequece A fuctio whose domai is N 2. Defie: a coverget sequece A sequece that has a limit 3. Is this sequece coverget? Why or why ot? l Yes, it is coverget sice L=0 by LHR.
INFINITE SERIES
Questio. Ca Joh ad Mark cross the ever-edig bridge?
Questio. Let x be the legth of the bridge. x x x x x 2 4 8 6 32... Note that this expressio ca also be writte as x 2
Questio 2. Ca we color the box completely?
Questio 2. 2 4 8 6 32... Note that this expressio ca also be writte as 2
Questio 3. Do you kow the sum? 2 4 8 6 32... Aswer:
Questio 4. Recall the sequece defied by. g 2 3 4 5 6 7 Now, what ca you say about...
Questio 4. WALANG SUM? MAY SUM -? 0? /2?? depede sa last term?
Questio 4.... NO, it does ot have a sum. Note that this expressio ca also be writte as
.3 INFINITE SERIES of CONSTANT TERMS
Defiitio. Let u be a sequece of real umbers ad s u u u u 2 3... The the sequece series. NOTATION: s s, is called a ifiite u
Defiitio. I a ifiite series, u u, u 2,..., u,... s, s 2,..., s,... s are called the terms of the ifiite series are called the partial sums of the ifiite series is the sequece of partial sums defiig the ifiite series
Example. Cosider 2 The first four terms of the series are u 2 u 2 3 4 u u 4 8 6 The first four partial sums of the series are s 7 s 3 2 2 4 8 8 3 s 2 2 4 4 5 s 4 2 4 8 6 6
Remarks: If s is the sequece of partial sums defiig the ifiite series The for 2, s s u. Our mai cocer o ifiite series is to determie whether the series coverges or ot. u
Defiitios. Cosider a ifiite series ad be the sequece of partial sums defiig the ifiite series. lim If exists ad is equal to, The: s u is coverget u S s S is the sum of the ifiite series
Defiitios. If lim s does ot exist, The: u is diverget ad it does ot have a sum.
Ca you add a ifiite umber of terms ad have a fiite sum?...... 2 2 4 8 2
Prove....... 2 2 4 8 2 PROOF. s... 2 4 8 2 2 s... 2 4 8 2 2 2 s s 2 2 2 s 2 2 2
PROOF. (cot.) Now, s 2 2 2 s 2 0 lim s lim 2 Thus, 2 is coverget ad its sum is.
Example 2. Cosider 3 32 Let u 332 Recall that 3 3 2 33 33 2
Example 2. (cot.) Now, 3 32 s u u u... u u 2 3 s 3 2 3 5 3 5 38 3 8 3... 3 3 4 3 3 3 3 3 3 2 A collapsig/telescopig series
Example 2. (cot.) So, s 3 2 3 3 2 0 lim s lim 6 3 3 2 6 Thus, 3 32 is coverget ad its sum is. 6
Theorem. (th term test) lim u 0 If u is coverget, the. PROOF. Suppose lim u is coverget with sum S. The. s S s s u Recall: Thus, lim u lim s s S S 0
Remark: (th term test for divergece) lim u 0 If, the u is diverget. BUT lim u 0 If, the u is either diverget or coverget.
Examples. Is the series diverget? 3 2 2 2 2 5 2 3 l 5 e si 2
Theorem. s If u is coverget ad is the sequece of partial sums defiig the series, the 0 for each, there exists a umber such R T that if ad are atural umbers such that N R N ad T N, the s s. R T
Theorem. The harmoic series is diverget....... 2 3 PROOF. Here we use the previous theorem with s s 2 R 2, T... 2 3...... 2 3 2
PROOF. (cot.) s 2 s... 2... 2... 2 2 2 2 That is, o umber s s 2 Thus, N exists such that whe. 2 is diverget.
Alterative Proof 2 2 2...... 2 3 2 3 4 4 2 4 5 6 7 8 8 2 8... 9 0 5 6 6 2 etc. Thus, k... 2 3 k 2 2 k lim diverges. k 2 is diverget.
Theorem. The geometric series ar a r a 0 a r r where ad are costats ad.. coverges to if ; 2. diverges if r. PROOF. 2 s a ar ar... ar 2 rs ar ar... ar ar s rs a ar
PROOF. (cot.) s r a r s rs a ar s If a r r a r lim s lim r r, the lim s. a r Now, If r, the lim u lim ar 0. Thus, theorem holds.
Examples. Determie if the geometric series is coverget. If it does, fid its sum. 4 5 2 3 4 3 3 2 7 2 5 2 e
Coverget Diverget u where lim s exists u where lim u 0 f f k k ar, r ar, r
Quiz No. 2 coverget or diverget? Briefly explai why.. 2. 3 5 3. 2 sec
4 Theorems about Ifiite Series. Cosider two ifiite series a ad b.. If they differ oly i a fiite umber of terms, the either both series coverge or both diverge.
4 Theorems about Ifiite Series. 2a. If the series a is coverget ad its sum S is, the the series ca is also coverget ad its sum is costat. c cs for each
4 Theorems about Ifiite Series. 2b. If the series a is diverget, the the series da ozero costat. is also diverget for each d
4 Theorems about Ifiite Series. 3. The sum or differece of two coverget series is also coverget. 4. The sum or differece of a coverget series ad a diverget series is diverget.
Examples. Determie whether the series is coverget or diverget. Explai why. 3 2 2 2 3 2 7 5 3 3 e
WARNING!!! The terms of a coverget series ca be grouped i ay way (provided that the order of the terms is maitaied), ad the ew series will coverge with the same sum as the origial series.
END