Slide 11.2.pptx - Convergent/Diverg ent Series Series vs...

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Convergent/Diverg ent Series
Series vs Sequences Sequence is a collection of numbers that are in one-to-one correspondence with the positive integers.: 0.3, 0.03, 0.003, 0.0003, etc. Series is an expression that represents an infinite sum of numbers: 0.3 + 0.03 + 0.003 + 0.0003 + etc. k-th partial sum S k of the series Σa n : S k = a 1 + a 2 + … + a n Sequence of partial sums of Σa n : S 1 , S 2 , S 3 , …, S n , ...
Series vs Sequences 0.3 + 0.03 + 0.003 + … S 1 = 0.3 S 2 = 0.33 S 3 = 0.333 Sequence of partial sums {S n } = 0.3, 0.33, 0.333, … Series Σa n converges if sequence {S n } converges : Conversely, Σa n diverges if sequence {S n } diverges. A divergent series has no sum
Example 1 Given the series a) Find S 1 , S 2 , S 3 , S 4 b) Find S n c) Does the series converge or diverge?
Example 2 Given the series a) Find S 1 , S 2 , S 3 , S 4 b) Find S n c) Does the series converge or diverge?
Harmonic Series Harmonic series: - The harmonic series is divergent {S n } diverges, so the series is divergent.
Geometric Series Geometric Series I. Converges and has the sum II. Diverges if We can verify this property in cases of r = 1, -1, 0.5, 2
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