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Given 1- cosx, how do you find the Taylor polynomial?

Answer 1

Evelyn Agard

We should know that:

#cosx=sum_(n=0)^oo(-1)^nx^(2n)/((2n)!)=1-x^2/(2!)+x^4/(4!)-x^6/(6!)+x^8/(8!)+...#

#1-cosx=1-(1-x^2/(2!)+x^4/(4!)-x^6/(6!)+x^8/(8!)+...)#

#color(white)(1-cosx)=x^2/(2!)-x^4/(4!)+x^6/(6!)-x^8/(8!)+...#

Noting this still alternates and that the factorials and powers increase by #2# in tandem:

#1-cosx=sum_(n=0)^oo(-1)^nx^(2n+2)/((2n+2)!)#

Which we could also write as:

#1-cosx=sum_(n=1)^oo(-1)^(n-1)x^(2n)/((2n)!)#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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