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First of all it says the remainder is:

f^(n+1)(c)(x-a)^(n+1)/(n+1)! for some c between a and x.

I do not know what c it means, how do I know what value to use as c?

My next issue is that for power series for sinx and cosx:

For example if I am doing sinx and use three terms x-x^3/3!+x^5/5! My previous derivative value that goes in front of the x^5/5! is 1 because cos(0)=1 but taylor's theorem states f^(n+1) so the next derivative would be -sin(0)=0. So would that mean the remainder theorem would be 0 in this case?

Something else I am confused about is what my book calls the "remainder estimation theorem" where it says if absolute value of f^(n+1)(t) is equal to or less than Mr^(n+1) then Rnx is = or < Mr^(n+1)(x-a)^(n+1)/(n+1)!

My book does 2 examples but it does not explain very well how to choose these values of M and r and how exactly I can use it to estimate.

I have read this chapter twice and either because of the way they word it or because I am thinking about it wrong I am unable to understand this remainder theorem which is frustrating because I feel good on the rest of the power series topic except for this. Thank you for any help.