How can I apply Taylor's theorem in this case?

[AObeid]

Hello, this is my first post on this forum! Glad I found a place that offers help on so many math subjects!
I've been having problems with one particular subject: Applying Taylor theorem to estimate an error.
Basically I have the function e^{(x^4)} . I was asked to find its power series so by using the template of e^x I found that e^{(x^4)}= 1 + x⁴ + x⁸/2! + x¹²/3! ...
so e^{(x^4)}=(summation) x⁴ⁿ/n! with n going from 0 till infinity.
Then I was asked to approximate the error with x = 0.5 while using P₆(x) or the 6th degree taylor polynomial.
Now I know that P₆(x) should be equal to 1+x⁴ = 1 +0.5⁴ since there is no 6th degree of this polynomial right?
ANd if that's correct then how can I find the expression for the remainder Rn₆(x) in order to estimate the error?
I know it should be the (n+1) derivative at a point c between 0 and x=0.5 but I couldn't go beyond that..
What am I missing here? Thanks in advance for any help!

 

[Ishuda]

Hello, this is my first post on this forum! Glad I found a place that offers help on so many math subjects!
I've been having problems with one particular subject: Applying Taylor theorem to estimate an error.
Basically I have the function e^{(x^4)} . I was asked to find its power series so by using the template of e^x I found that e^{(x^4)}= 1 + x⁴ + x⁸/2! + x¹²/3! ...
so e^{(x^4)}=(summation) x⁴ⁿ/n! with n going from 0 till infinity.
Then I was asked to approximate the error with x = 0.5 while using P₆(x) or the 6th degree taylor polynomial.
Now I know that P₆(x) should be equal to 1+x⁴ = 1 +0.5⁴ since there is no 6th degree of this polynomial right?
ANd if that's correct then how can I find the expression for the remainder Rn₆(x) in order to estimate the error?
I know it should be the (n+1) derivative at a point c between 0 and x=0.5 but I couldn't go beyond that..
What am I missing here? Thanks in advance for any help!

Although the template can find the coefficients for the series, you will need the actual derivative for the error term. So, letting t=x⁴ for convenience we have
f⁽⁰⁾(x) = e^t; f⁽⁰⁾(0) = 1
f⁽¹⁾(x) = 4x³ e^t; f⁽¹⁾(0) = 0
f⁽²⁾(x) = ...; f⁽²⁾(0) = 0
f⁽³⁾(x) = ...; f⁽³⁾(0) = 0
f⁽⁴⁾(x) = ...; f⁽⁴⁾(0) = 24
f⁽⁵⁾(x) = ...; f⁽⁵⁾(c) = ...
and find the c which maximizes f⁽⁵⁾(c) for c between 0 and 0.5.