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Estimate number of terms needed for taylor polynomial

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1. Homework Statement
For ln(.8) estimate the number of terms needed in a Taylor polynomial to guarantee an accuracy of 10^-10 using the Taylor inequality theorem.

2. Homework Equations
|Rn(x)|<[M(|x-a|)^n+1]/(n+1)! for |x-a|<d.

3. The Attempt at a Solution
All I've done so far is take a couple derivatives of ln(x):
1/x
-1/x^2
2!/x^3
-3!/x^4
and made a general (-1)^(n+1) (n-1)!/(x^n)
At this point I am lost, am I supposed to assume a=0? Then M, which is the absolute value of the n+1 derivative would be n!/.8^(n+1). And plugging that in to the Taylor inequality equation the .8^(n+1) would cancel and so would the n! and I would be left with 1/n+1<10^-10, which gives me a really big number, even though the answer should be 14. What am I missing?

EDIT: It looks like there might be some missing information. My professor sent the solution out, and to me it looks like he is using a=1. Can someone confirm that?
FyyYfr1.png
 
Last edited:
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9,202
The relevant equation is missing an important part: "for all a (that satisfy the second inequality)". You have to find the value of "a" that maximized the uppper limit to find the true upper limit.
Also, did you find d?
 

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