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 

|Tn(x)−ln(1−x)|<0.01

 for all x in the interval 

(-.5, .5)


Tutor Answer

(Top Tutor) Daniel C.
(997)
School: UIUC
PREMIUM TUTOR

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) = − ∑
𝑘=1

𝑥𝑘
.
𝑘

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

𝑓 (𝑛+1) (𝜉) 𝑛+1
𝑥
,
(𝑛 + 1)!

wh...

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