Analysis 2 Notes 4 Flashcards
(4O-N28-) What does power series converge imply about …What does absolute convergence imply
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(4D-N29-4.1) Definition of radius of convergence of a power series.
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(4E-N29-4.1) Radius of convergence of power series x^n
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(4E-N30-4.2) Radius of convergence of power series x^n/ n!
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(4E-N30-4.3) Radius of convergence of power series n! x^n
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(4T-N30-4.1) If |x| < R then the power series converges absolutely and if |x| > R then the power series diverges
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(4L-N31-4.1) PS An x^n from n =0 up, and nAn x^n from n = 1 have the same radius of convergence
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(4L-N32-4.2) nanx^(n−1) and An x^n have the same radius of convergence
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(4L-N32-4.3) An x^n and n!/(n-k)! an x^(n-k) have the same radius of convergence
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(4L-N32-4.4) |((xo + h)^n - xo)/h - nxo ^(n-1)| less than of equal to (n(n-1)|h| / 2)(|xo| + |h|)^n-2
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(4T-N33-4.2) Let IS Anx^n be a power series with radius of convergence R > 0 and let f : (−R,R) → R be the sum of this series. Then f is differentiable on the interval (−R,R) and f′(x) = IS nAn x^(n-1)
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(4C-N34-3.1) Let IS Anx^n be a power series with radius of convergence R > 0 and let f : (−R,R) → R be the sum of this series. Then f is infinitely differentiable on the interval (−R,R) and f(k)(x) = IS nAn x^(n-k)
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(4C-N34-3.2) Let IS Anx^n be a power series with radius of convergence R > 0 and let f : (−R,R) → R be the sum of this series. Then an = f(n) (0) / n!
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(4R-N34-) Not every infinitely differentiable function can be written as a power series
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(4T-N35-4.3) Let IS Anx^n be a power series with radius of convergence R > 0. Then integral 0 to x of ( IS Ant^n) dt = IS An (x^(n+1)/n+1)
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