Math118_S10_W8

Math118_S10_W8 - Math118 (M. Kohandel) -Outline (weeks 8...

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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)
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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.
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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
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Math118_S10_W8 - Math118 (M. Kohandel) -Outline (weeks 8...

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