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Using power series remainder term

  • Thread starter morenogabr
  • Start date
  • #1
28
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Homework Statement


(For power series about x=1) Using the error formula, show that [tex]\left|ln(1.5)-p_{3}(1.5)\right|\leq\frac{(0.5)^{4}}{4} [/tex]


Homework Equations


[tex]p_{3}(x) = x-1 - \frac{(x-1)^{2}}{2} + \frac{(x-1)^{3}}{3}[/tex]
[tex]\\\epsilon_{n}(x)=\frac{f^{n+1}(\xi)}{(n+1)!}(x-x_{o})^{n+1}\\where \xi lies between x_{o} and x[/tex]


The Attempt at a Solution


Im sure this is an easy one, but I cant think of any useful relationship between the difference |f(x)-p_3(x)| and that piece of the remainder function... any hints?
(excuse my latex crappiness)
 

Answers and Replies

  • #2
58
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Isn't there supposed to be a sigma notation (for sum) on the left of the first term of the second equation (power series)?
 
  • #3
28
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No, its a 3rd order polynomial that approximates the power series, and its written out term by term so sigma is uneeded. Oh, if you mean for the epsilon, its the remainder term used to measure accuracy of the approximation. Its a single term, so no sigma needed. I guess I should say something like n=3 (bc of p_3(x)) so epsi_4(x)...
 

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