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.

$$\sum_{n=0}^\infty f^n(c)\frac{(x-c)^n}{n!}$$ 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.

$$\sum_{n=0}^\infty f^n(c)\frac{(x-c)^n}{n!}$$ 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)= $$\frac{(1)}{(1+x)^4}$$
f¹(x)= $$\frac{(-4)}{(1+x)^5}$$
f²(x)= $$\frac{(20)}{(1+x)^6}$$
f³(x)= $$\frac{(-120)}{(1+x)^7}$$
f⁴(x)= $$\frac{(840)}{(1+x)^8}$$

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

 

[PeroK]

f⁰(x)=(1+x)^-4
f¹(x)= [itex]\frac{-4}{(1+x)⁵} [\itex]<br /> f²(x)= [itex]\frac{20}{(1+x)⁶} [\itex]<br /> f³(x)= [itex]\frac{-120}{(1+x)⁷} [\itex]<br /> f⁴(x)= [itex]\frac{840}{(1+x)⁸} [\itex]<br /> <br /> I think that I got the Taylor Series of $$fⁿ(x)=\frac{(-1)ⁿ⁺¹*(n-1)!}{2ⁿ⁺⁴}$$[/itex][/itex][/itex][/itex]

[itex][itex][itex]$<br /> Can you fix that latex? In any case, it doesn't look right. It looks like a mixture of parts of the solution.$[/itex][/itex][/itex]

 

[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 $$f^n(x)=\frac{(-1)^(n)*\frac{(n-1)!}{6}}{(1+x)^(n+4)}$$

 

[PeroK]

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

 

[deagledoubleg]

Does the above look better?