# Lecture 19 - Review of Power Series and Taylor Polynomials...

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Review of Power Series and Taylor Polynomials

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Infinite Sums and Power Series Recall Infinite Sums: 1 2 + 1 4 + 1 8 + 1 16 + ... + 1 2 n +1 + ... 1 2 0+1 + 1 2 1+1 + 1 2 2+1 + 1 2 3+1 + ... + 1 2 n +1 + ...
Infinite Sums and Power Series Recall Infinite Sums: 1 2 0+1 + 1 2 1+1 + 1 2 2+1 + 1 2 3+1 + ... + 1 2 n +1 + ... 1 X n =0 1 2 n +1 =

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Infinite Sums and Power Series Recall Infinite Sums: 1 X n =0 1 2 n +1 In General: 1 X n =0 b n b n = 1 2 n +1
Infinite Sums and Power Series In General: 1 X n =0 b n Three possible outcomes of infinite sums: = 1 C = 1 or C Diverges Converges Neither 1 X n =0 b n 1 X n =0 b n 1 X n =0 b n =

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Infinite Sums and Power Series In General: 1 X n =0 b n Special Type of Infinite Sum: Power Series b n = a n ( x - x 0 ) n a n ( x - x 0 ) n Which gives 1 X n =0
Infinite Sums and Power Series In General: 1 X n =0 b n Special Type of Infinite Sum: Power Series a n ( x - x 0 ) n 1 X n =0 Depends on x

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Infinite Sums and Power Series In General: 1 X n =0 b n Special Type of Infinite Sum: Power Series a n ( x - x 0 ) n 1 X n =0 Depends on x = f ( x )
Infinite Sums and Power Series In General: 1 X n =0 b n Special Type of Infinite Sum: Power Series a n ( x - x 0 ) n 1 X n =0 = f ( x ) Depending on x , can either diverge , converge , or neither

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