Find Taylor Series from a function and its interval of convergence

[deagledoubleg]

Let f(x) = (1+x)^-4
Find the Taylor Series of f centered at x=1 and its interval of convergence.

[tex] \sum_{n=0}^\infty f^n(c)\frac{(x-c)^n}{n!}[/tex] is general Taylor series form

My attempt

I found the first 4 derivatives of f(x) and their values at fⁿ(1). Yet from here I do not know how to find the taylor series, from there I should be able to finish it myself. Any ideas on how to find the Taylor series here?

 

[PeroK]

Let f(x) = (1+x)^-4
Find the Taylor Series of f centered at x=1 and its interval of convergence.

[tex] \sum_{n=0}^\infty f^n(c)\frac{(x-c)^n}{n!}[/tex] is general Taylor series form

My attempt

I found the first 4 derivatives of f(x) and their values at fⁿ(1). Yet from here I do not know how to find the taylor series, from there I should be able to finish it myself. Any ideas on how to find the Taylor series here?

How did you find the derivatives?

 

[deagledoubleg]

f⁰(x)= [tex] \frac{(1)}{(1+x)^4}[/tex]
f¹(x)= [tex] \frac{(-4)}{(1+x)^5}[/tex]
f²(x)= [tex] \frac{(20)}{(1+x)^6}[/tex]
f³(x)= [tex] \frac{(-120)}{(1+x)^7}[/tex]
f⁴(x)= [tex] \frac{(840)}{(1+x)^8}[/tex]

I think that I got the Taylor Series of [tex] f^n(x)=\frac{(-1)^(n+1)*(n-1)!}{2^(n+4)} [/tex]

 

[PeroK]

f⁰(x)=(1+x)^-4
f¹(x)= [itex] \frac{-4}{(1+x)⁵} [\itex]
f²(x)= [itex] \frac{20}{(1+x)⁶} [\itex]
f³(x)= [itex] \frac{-120}{(1+x)⁷} [\itex]
f⁴(x)= [itex] \frac{840}{(1+x)⁸} [\itex]

I think that I got the Taylor Series of [tex] fⁿ(x)=\frac{(-1)ⁿ⁺¹*(n-1)!}{2ⁿ⁺⁴} [/tex]

Can you fix that latex? In any case, it doesn't look right. It looks like a mixture of parts of the solution.

 

[deagledoubleg]

Fixed

 

[PeroK]

Fixed

You can see that your general formula doesn't match the first four derivatives.

 

[deagledoubleg]

True, I can see that now. I think the issue is the (n-1)! in the numerator. I can't find something that works because I want it to be n! but to stop at 4. like 7*6*5*4, but not include *3*2*1. To do this would I just divide that by 6?

 

[PeroK]

True, I can see that now. I think the issue is the (n-1)! in the numerator. I can't find something that works because I want it to be n! but to stop at 4. like 7*6*5*4, but not include *3*2*1. To do this would I just divide that by 6?

For the time being, just calculate ##f^{(n)}(x)##. You're not aiming at anything.

And, yes, ##7*6*5*4 = 7!/6##

 

[deagledoubleg]

I think that I have [tex] f^n(x)=\frac{(-1)^(n)*\frac{(n-1)!}{6}}{(1+x)^(n+4)} [/tex]

 

[PeroK]

You need to check that ##(n-1)!##.

 

[deagledoubleg]

Does the above look better?