hw-11-09-func-as-power-series

hw-11-09-func-as-power-series - * letting a=n in sum a x^n;...

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Chapter 11.9 homework: Representations of Functions as Power Series I strongly, strongly recommend that you do each problem on its own page-- give yourself a lot of room! Keep everything in neat columns! #32 Show that f(x) = sum{n=0 to infinity} (-1)^n x^(2n)/(2n)! solves f''(x) + f(x) = 0 #33 Bessel function of order 0 #34 Bessel function of order 1 #35 show that f(x) = sum{n=0 to infinity} (x^n) / n! solves f'(x) = f(x) #38 using the geometric series, find sum{n=1 to infinity} n* x^(n-1) Question A: Visit the various "Bessel Function Demo" links. (mandatory!) Question B: (optional) Revisit Chapter 7.5 problem #81: find the antiderivative of (2x^2+1)*exp(x^2) dx Common Problems:
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Unformatted text preview: * letting a=n in sum a x^n; a can't depend on n * moving n out of a sum: sum n x^n cannot become n * sum x^n * ending up with an n in the final answer, similar to a definite integral of f(x) dx; shouldn't have an x in answer * not estimating/doing LB/UB: try first few terms of sum to see the general value you should end up with. * On the other hand, don't rely solely on partial sum calculations! For example, what if your partial sums are 0.0769 + 0.0386 + 0.0256 + 0.0192 + 0.0154 + . .. does that converge? You can't tell just by looking! (these terms happen to be 1/(13n) , so will the sum converge? no, it's (1/13) times the Harmonic Series)...
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This note was uploaded on 09/08/2011 for the course MATH 121 at Eastern Michigan University.

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