Sigma Notation We use to indicate the sum am+am+l+.…+anr For instance, a3+a4+a5+a6+a7. Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Sigma Notation We use to indicate the sum am + am+1 + · · · + an . For instance, a3 + a4 + a5 + a6 + a7 . n k k m a = 7 7 7 3 3 3 , , i j k i j k a a a = = =
Infinite Series Definition: Let aa,az....be an infinite sequence ofreal numbers. We can form the partial sums 0=4w=2a6 S=a0+a1=】 s=a+a,+a= ak 气=a+4+4+a,=w =a,+a+0,+a++a,=2☑ and so on Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Infinite Series Definition: Let a0 , a1 , a2 , . . . . be an infinite sequence of real numbers. We can form the partial sums: 0 0 0 0 1 1 0 1 0 2 2 0 1 2 0 3 3 0 1 2 3 0 0 1 2 3 0 , , , , and so on. k k k k k k k k n n n k k s a a s a a a s a a a a s a a a a a s a a a a a a = = = = = = = = + = = + + = = + + + = = + + + + + =
Infinite Series DEFINITION 12.2.1 If,as n->oo,the sequence of partial sums =2 =0 tends to a finite limit L,we write and say that the series converges to L We call L the sum of the series.If the sequence of partial sums diverges,we say that the series diverges Main Menu cew时
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Infinite Series
Infinite Series Example 豆-r 22 Main Menu 0
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Infinite Series Example = − 0 ( 1) k k =0 2 k k
Infinite Series The Geometric Series The series 1,x,x2,x3,... is called a geometric progression The sum of these numbers is called the geometric series: -0 (① If x|<1, then xk 1-x (12.2.2) (ii） If x≥1， then 0 Main Meny c的天
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Infinite Series The Geometric Series The series 1, x, x2 , x3 , . . . is called a geometric progression The sum of these numbers is called the geometric series: 0 k k x =
Infinite Series (12.2.3) 15 32 S2 45S5 Figure 12.2.1 Main Meny 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Infinite Series
Example:A ball dropped from height h hits the floor and rebounds to a height proportional to h,that is,to a height oh with<1.It then falls from height oh,hits the floor and rebounds to the height o(oh)=oh, and so on.Find the length of the path of the ball. Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example: A ball dropped from height h hits the floor and rebounds to a height proportional to h, that is , to a height σh with σ<1. It then falls from height σh, hits the floor and rebounds to the height σ(σh)= σ 2h, and so on. Find the length of the path of the ball
Infinite Series THEOREM 12.2.4 1.If comverges and converges,thenb)converges. 0 k=0 k=0 Morever,if ak=Land∑bs=M,then∑(a+b)=L+M k=0 =0 k=0 2.If converges,then converges for each real mumber a. k=0 k=0 Moreover,if 4=L,hen k=0 k-0 Main Meny c的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Infinite Series
Infinite Series THEOREM 12.2.5 The kth term of a convergent series tends to 0;namely, converges, then ak→0 as k→00 0 [Main Menu☐ 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Infinite Series
(12.2.6) if ak≯0 then diverges. Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved