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## Main Question or Discussion Point

I have a few questions about the remainder theorem.

1: For series that "skip" terms (example: 1+x^2+x^4+x^6) the theorem says the n+1 derivative and x^(n+1)/(n+1)!. For example if you have 1 + x^2 where you know the next term would be x^4 you could treat it as a third order or a second order. My book has situations where they do both variations and it is extremely confusing because I do not know which to choose.

Example: This is a problem I made up: Estimate error involved in using 1+x^2 + (x^4)/2 to estimate e^(x^2) on -0.1 to 0.1.

So do I treat it as forth order or fifth order? Because my book has situations similar to this where it has done both so I have no idea what to do? It asks for n+1 derivative so is it the fifth or six derivative? Do I do (.1)^6/6! or (.1)^5/5!.

This is a major issue I am having please explain why it is one or the other because I can see why it could go either way.

2: The best way to ask my second question is to just show another example it is difficult to explain.

Example: Use x-(x^2)/2 to estimate ln(x+1) over the interval [0, 0.2].

The theorem states f^(n+1)(c) (x)^(n+1) / (n+1)!

Here is what I believe: The c value and the x value can be different in order to maximize the different parts. Is this true?

For this example I would have c be 0 in order to maximize the third derivative of ln(x+1) which is 2/(x+1)^3.....but I would have x be .2 in order to maximize the (x)^3 part of the theorem?

Is what I assumed correct? You can have different values of c and x in order to maximize the different parts? I believe this is correct so hopefully you can confirm this, if not can you please explain why?

Thank you very much for replies, both of these questions have been confusing to me and any help is appreciated

1: For series that "skip" terms (example: 1+x^2+x^4+x^6) the theorem says the n+1 derivative and x^(n+1)/(n+1)!. For example if you have 1 + x^2 where you know the next term would be x^4 you could treat it as a third order or a second order. My book has situations where they do both variations and it is extremely confusing because I do not know which to choose.

Example: This is a problem I made up: Estimate error involved in using 1+x^2 + (x^4)/2 to estimate e^(x^2) on -0.1 to 0.1.

So do I treat it as forth order or fifth order? Because my book has situations similar to this where it has done both so I have no idea what to do? It asks for n+1 derivative so is it the fifth or six derivative? Do I do (.1)^6/6! or (.1)^5/5!.

This is a major issue I am having please explain why it is one or the other because I can see why it could go either way.

2: The best way to ask my second question is to just show another example it is difficult to explain.

Example: Use x-(x^2)/2 to estimate ln(x+1) over the interval [0, 0.2].

The theorem states f^(n+1)(c) (x)^(n+1) / (n+1)!

Here is what I believe: The c value and the x value can be different in order to maximize the different parts. Is this true?

For this example I would have c be 0 in order to maximize the third derivative of ln(x+1) which is 2/(x+1)^3.....but I would have x be .2 in order to maximize the (x)^3 part of the theorem?

Is what I assumed correct? You can have different values of c and x in order to maximize the different parts? I believe this is correct so hopefully you can confirm this, if not can you please explain why?

Thank you very much for replies, both of these questions have been confusing to me and any help is appreciated