Question

what is the remainder of a taylor series?

Answer 1

it tends to be the average of the place you stopped at and the next value calculation; having to do with integration

Answer 2

\[\sum_{1}^{\infty}f(n)x^n=S_n+\frac{\int R_n+\int R_{n+1}}{2}\]

Answer 3

or is it half the difference?

Answer 4

im thinking its along the lines of
\[\sum_0x^n=x^0+x^1+x^2+x^3+R_4\]

where the remainder cam be approximated with integration

Answer 5

Actually I can't visualize it...

Answer 6

is there a particular taylor you need to assess? or is this just a general query?

Answer 7

Yes I need to give the taylor series of 3rdroot of x

Answer 8

the 4 first terms

Answer 9

and give the remainder after the 4th term

Answer 10

cube root of x?

Answer 11

have you determined the taylor for it yet?

Answer 12

yes

Answer 13

\[f(x)=x^{1/3}\\
f'(x)=\frac13x^{-2/3}\\
f''(x)=-\frac29x^{-5/3}\\
f''(x)=\frac{10}{27}x^{-7/3}\\
\]
at x=1 these all form the coeffs of the taylor
\[\frac{1}{1},\frac13,-\frac{1(2)}{9},\frac{1(2)(5)}{27},-\frac{1(2)(5)(7)}{36}\]
what did you get for the taylor?

Answer 14

\[f \prime \prime \prime \prime (x)=\frac{ 10 }{ 27}x ^{-8/3}\]

Answer 15

otherwise i get the same for the derivatives

Answer 16

and I don't substitute one into x... i keep the x like that

Answer 17

you have to sub in a value to x (say x=a) to define the coeffs of the taylor series with a variable (x-a)^n

Answer 18

might not have that exactly recalled, but the derivatives define a sequence for the coeffs of the taylor

Answer 19

the series representation of cbrt(x) applies a binomial calculation ... 1/3 C n