Find number of terms in taylor expansion upto when error is 10^(-6)

Find number of terms in taylor expansion upto when error is 10^(-6)

[Walter Roberson]

I don't think there is any closed formula for this for arbitrary input functions. maybe there is some kind of formula involving fractional integration, but you would not be able to find a closed form integral for most functions.

[James Tursa]

Did you have a specific function in mind?

[khadija raza]

yes let me elaborate. Find number of terms in the expansion (-1)^(n+1)/(2n-1) upto when error in the expansion is within (10^(-6))

[khadija raza]

plz help me as soon as possible plz

[Walter Roberson]

If n can be fractional, then this involves complex numbers.

The number of terms required for expansion is going to depend upon the magnitude of n.

[khadija raza]

what about error sir Walter? we have to consider the error to find number of terms

[Walter Roberson]

The number of terms is going to be quite different for n around 5 than it is for n around 555555555555555555555555555 . The magnitude of n matters because your error requirement is stated in absolute form, not in relative form.

I suspect the taylor series is only valid for continuous functions, but if you do not restrict n to integer then you can get large complex magnitudes.

As you have not stated an expansion point, it would make more sense to use a series expansion instead of a taylor expansion, or at least a Maclaurin series (taylor around 0)

Answer 1

[Aquatris]

There is an approximation for upper bound on the error (or remaining terms in Taylor expansion). Here is a link that explains the method. From there, it is still not a closed form solution but you can roughly guess the order you want for desired error.

[Walter Roberson]

khadija raza comments,

i posted the wrong question with wrong tag