Taylor's inequality,math homework help

Dec 5th, 2016
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Question description

Taylor series of ln(1-x)

Find the smallest value of n such that Taylor's inequality guarantees that 


 for all x in the interval 

(-.5, .5)

Tutor Answer

(Top Tutor) Borys S
School: UT Austin

This is the solution using the Lagrange remainder and its straightforward estimate. Tell me if this is not what you want.

The Taylor series centered at 𝑥 = 0 for the function 𝑓(𝑥) = ln(1 − 𝑥) is

𝑇(𝑥) = − ∑


The remainder of the Taylor 𝑛-th partial sum may be expressed as
𝑓(𝑥) − 𝑇𝑛 (𝑥) =

𝑓 (𝑛+1) (𝜉) 𝑛+1
(𝑛 + 1)!


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