Expert: Paul Klarreich - 11/8/2008

Question
I am B.Sc first year student and it comes from the section differential calculus.
The question goes like this-
Expand sin x by validity power series expansion.
I can easily expand it by using maclaurin theorem,but I just could not understand what does the valdity power series expansion means.
In the book the answer has been done by using lagrange's form of remainder,but I just couldnot understand it.
Please explain it thoroughly to me.If there are any links for me,pleae post it as well.

Answer
Questioner:   Roshan
Category:  Calculus
Private:  No
 
Subject:  Maclaurin series
Question:  I am B.Sc first year student and it comes from the section differential calculus.
The question goes like this-
Expand sin x by validity power series expansion.
I can easily expand it by using maclaurin theorem,but I just could not understand what does the valdity power series expansion means.
In the book the answer has been done by using lagrange's form of remainder,but I just couldnot understand it.
Please explain it thoroughly to me.If there are any links for me,pleae post it as well.
.................................
Hi, Roshan,

It's the Taylor series expansion that you want to study.  See: Taylor's theorem with remainder, and the proof that the remainder term approaches zero.  (Maclaurin series is just Taylor series with  a=0.)  For a complete explanation, you will have to look in the text or ask your teacher -- it's too much for just an 'answer.'

There are several places on the web where you can find a proof of this.  As you might expect, it uses the MVT (or Rolle's Theorem)