Section10.6-Power Series

Section10.6-Power Series - Section10.6 PowerSeries Example...

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Section 10.6 Power Series
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Example 23 0 1 nn n xx x x x   This is both a power series and a geometric series. Convergent for | x | < 1 Interval of convergence: ( 1 , 1 ) note Radius of convergence, R = 1
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In General Power Series in (x–c) Power Series centered at c Power Series about c  2 01 2 0    nn n bxc b b xc For 23 0 1 n xx x x x  0; 1  n cb (for all n)
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Example 23 0 1 !2 ! 3 ! n n xx x x n  Find the radius and interval of convergence. 1 0; !  n cb n Use the Ratio Test:  1 1 1 ! 1! ! 1 1 n n n n n n x a n an x x n n x x n    
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Example continued 1 1 1 n n a x an    1 lim 0 n n n a a  for all x Interval of convergence: ( ‐ ∞ , ) Radius of convergence, R =
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Example  23 0 !31 3 2 !3 3 n n nx x x x   1 1 1! 3 13 n n n n a a   1 lim n n n a a   unless | x–3 | = 0 Interval of convergence: x = 3 Radius of convergence, R = 0 Find the radius and interval of convergence:
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Example  0 32 1 n n x n Find the interval and radius of convergence: 1 1 1 1 2 1 2 1 2 2 n n n n n n x a n an x x n n x n x n   1 lim 2 n n n a x a 
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Example continued 1 lim 2 n n n a x a   Convergent for | x–2 | < 1 Divergent for | x–2 | > 1 What about | x–2 | = 1 ? Ratio Test is inconclusive. Use another test.
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Example – continued  0 32 1 n n x n 0 21 3 1 n x n 0 31 1 n n x n  Checking | x–2 | = 1 Divergent Convergent Series is convergent for 1 x–2 < 1 ; 1 x < 3 Interval of convergence: [ 1, 3 ) Radius of convergence, R = 1
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Theorem For a given power series, , there are only three possibilities: i) The series converges only when x = c.
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Section10.6-Power Series - Section10.6 PowerSeries Example...

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