{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Math118_S10_W8

# Math118_S10_W8 - Math118(M Kohandel-Outline(weeks 8 9 8.1...

This preview shows pages 1–4. Sign up to view the full content.

Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 1 Outline (weeks 8 & 9) 8.1 Infinite series of numbers (10.9) 9.1 Integral, comparison, and limit comparison tests (10.10) 9.2 Limit ratio and limit root tests (10.11)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 2 8.1 Infinite sequences of numbers We saw that for a power series of the form 0 n n n x a , the open interval of convergence can be determined from | / | lim 1  n n n a a R . We then say that the power series is convergent for R x R . What happens at R x ? For example, 2 | 2 / ) 1 ( 2 / ) 1 ( | lim | | lim or 2 2 1 2 / 1 , 2 2 ) 2 / ( 1 1 ) 2 ( 2 ) 1 ( 1 1 1 0 0   n n n n n n n n n n n n n n a a R x x x x x x At 2 x , we have 0 ) 1 ( n n and at 2 x the power series reduces to 0 1 n . Both of these series are divergent - so the power series is convergent on 2 2 x . In general, when specific values of x are substituted into power series, we obtain "infinite series of numbers". For example,   1 1 1 1 1 3 ) 1 ( 3 | ) 3 ) 1 /(( ) 1 ( ) 3 /( ) 1 ( | lim , 3 ) 1 ( n n n x n n n n n n n n n n n n R x n Infinite series of numbers (or just series) also arises when we consider an infinite sequence of numbers, } { n a , and write 1 n n a . For example, 2 2 2 1 2 2 4 3 3 2 2 1 ) 1 ( } ) 1 ( { n n n n n We denote by n S the sum of the first n terms of the series 1 n n a , then n k k n n a a a a S a a S a S 1 2 1 2 1 2 1 1 , , , is called the "sequence of partial sums" of the series 1 n n a . Definition: Let n k k n a S 1 be the n th partial sum of a series 1 n n a . If the sequence of partial sums } { n S has limit n n S S lim (it is convergent), then the sum of the series is S a n n 1 . If } { n S does not have a limit, we say that the series does not have a sum.
Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 3 Example 8.1: Geometrical series 7.2).] example (see exist not does limit the otherwise , 1 | | if 0 lim : [Note 1 | | : 1 lim 1 - 1 lim 1 ) 1 ( 1 ) ( diverges as 1 ) ( ) 0 ( Subst. 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 8

Math118_S10_W8 - Math118(M Kohandel-Outline(weeks 8 9 8.1...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online