Lagrange Remainder for Taylor Expansion of ln(4/5) ≤ 1/1000?

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SUMMARY

The discussion centers on determining the Lagrange remainder for the Taylor expansion of ln(4/5) to ensure it is less than or equal to 1/1000. The user attempted to apply Taylor series using both ln(1+x) with x=-1/5 and ln(x) with x=4/5, but consistently derived that n must be at least 3. However, upon further analysis, it was concluded that n must actually be 4 to meet the error requirement, indicating a misunderstanding in the application of the remainder formula.

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peripatein
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Hi,

Homework Statement


I am trying to limit Lagrange's remainder on taylor expansion of ln(4/5) to be ≤ 1/1000.


Homework Equations





The Attempt at a Solution


I have tried using both ln(1+x), where x=-1/5 and x0(the center)=0, and ln(x), where x=4/5 and x0=1.
Every time I keep getting that (n+1)4n+1≥1000, leading to n ≥ 3.
But then, upon expansion up to the third power, I keep getting a result whose error is greater than the desired 1/1000. It appears the result should have been n≥4, but why so when algebra seems to prove it not to be the case?
I'd appreciate some advice. Obviously I am missing something.
 
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I am rather surprised no one has replied. Is there anything amiss with my formulation of the problem?
 
peripatein said:
Every time I keep getting that (n+1)4n+1≥1000, leading to n ≥ 3.
Please post your working to that point.
 

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