Calc09_1 - 9.1 9.1 Power Series Power Series Greg Kelly...

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Unformatted text preview: 9.1 9.1 Power Series Power Series Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2003 This is an example of an This is an example of an infinite series infinite series . . 1 1 Start with a square one Start with a square one unit by one unit: unit by one unit: 1 2 1 2 1 4 1 4 + 1 8 1 8 + 1 16 1 16 + 1 32 1 64 1 32 + 1 64 + 1 + ⋅⋅⋅ = This series This series converges converges (approaches a limiting value.) (approaches a limiting value.) Many series do not converge: Many series do not converge: 1 1 1 1 1 1 2 3 4 5 + + + + + ⋅⋅⋅ = ∞ → In an infinite series: In an infinite series: 1 2 3 1 n k k a a a a a ∞ = + + + ⋅⋅⋅ + + ⋅⋅⋅ = ∑ a a 1 , a , a 2 ,… ,… are are terms terms of the series. of the series. a a n is the is the n n th th term term . . Partial sums: Partial sums: 1 1 S a = 2 1 2 S a a = + 3 1 2 3 S a a a = + + 1 n n k k S a = = ∑ n n th th partial sum partial sum If If S S n has a limit as , then the series converges, has a limit as , then the series converges, otherwise it otherwise it diverges diverges ....
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