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Definition
|Rn (x)| ≤ M/n+1 |x-x0|n+1
Where M = upper bound |
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| What series is = ∑x2 as x goes to infinity. |
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Definition
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| Theorem: For a power series ∑ck(x-x0)k, exactly one of the fallowing statements is true |
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Definition
a) The series converges only for x=x0
b) The series converges absolutely for all real values of x
c) Th series converges absolutely for all x in some finite open interval is (x0-R,x0+R) and diverges if x<x0-R or x>x0+R. At either of the values x=x0-R or x=x0+R, the series may converge absolutely, converge conditionally, or diverge, depending on the particular series. |
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Theorem: For any power series in x, exactly one of the following is true:
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Definition
a) the series converges for x=o
b) the series converges absolutely for all real values of x
c) the series converges absolutely for all x in some finite open interval (-R,R) and diverges if x<-R or x>R. At either of the values x=R or x=-R, the series may converge absolutely, converge conditionally, or diverge depending on the particular series. |
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Definition
| ∑ f(k)(X0)/K! = f(x) + f'(x)(x-x0) + f''(x)(x-x0)/2! |
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Definition
| If the limit of Uk doesn't = o the series of Uk diverges. |
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