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

[peripatein]

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.

 

[peripatein]

I am rather surprised no one has replied. Is there anything amiss with my formulation of the problem?

 

[haruspex]

Every time I keep getting that (n+1)4ⁿ⁺¹≥1000, leading to n ≥ 3.

Please post your working to that point.