MTH 251 Lecture Notes - Lecture 32: Squeeze Theorem, Ratio Test

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Lecture Notes
Tn is the  degree Taylor polynomial of at .
Setting  where  is called the remainder of the Taylor series
If we can show that 

Then we have: 

 

Theorem:
If
Where is the nth degree Taylor polynomial and 
 then is the sum of the
Taylor series on the internal  where is the radius of convergence and a is the
center.
Two useful things
Taylor’s Inequality
 for  then the 
 for
Also, 

 for all
Back to :
If then is

 For all
Then,  for
Then, by Taylor’s inequality,

 For
Now notice that,


 By squeeze theorem,
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