I How does this binomial expand?

[lioric]

TL;DR Summary

I’m trying to expand 1/(1+𝑥)^2 and (1+𝑥)^-2

1/(1+x)² and (1+x)^-2 are the same but the negative index binomial has an infinite expansion, but the other as a denominator, stops at x ²

So please help me understand this
Thank you in advance.

 

[PeroK]

TL;DR Summary: I’m trying to expand 1/(1+𝑥)^2 and (1+𝑥)^-2

1/(1+x)² and (1+x)^-2 are the same but the negative index binomial has an infinite expansion, but the other as a denominator, stops at x ²

So please help me understand this
Thank you in advance.

$$\frac 1 {(1+x)^2} = \frac 1 {1 + 2x + x^2}$$That is not a power series in $x$. That is the reciprocal of quadratic in $x$. That function can also be expressed, using the Binomial Theorem, as an infinite power series in $x$:
$$\frac 1 {(1+x)^2} = (1 + x)^{-2} = 1 -2x + 3x^2 - 4x^3 \dots$$Note, however, that this power series has a radius of convergence and is not valid for all $x \in \mathbb R$.

 

[lioric]

Yes I know that. But what’s bugging me is, what if I do the reciprocal after expanding the bracket, them it’s not the same.
If I expand the (1+x)² and i do a reciprocal,(which I still don’t know how) I don’t see how I could get the 1-2x+3x²-4x³…….

 

[PeroK]

Yes I know that. But what’s bugging me is, what if I do the reciprocal after expanding the bracket, them it’s not the same.
If I expand the (1+x)² and i do a reciprocal,(which I still don’t know how) I don’t see how I could get the 1-2x+3x²-4x³…….

$$(1 + (2x + x^2))^{-1} = 1 - (2x + x^2) + (2x + x^2)^2 - (2x + x^2)^3 \dots$$$$= 1 - 2x + 3x^2 -4x^3 \dots$$

 

[lioric]

Thank you for this. Could you explain a bit more how this form is derived? Im a bit unfamiliar with the (1+(2x+x²))^-1 part came from.

 

[PeroK]

Thank you for this. Could you explain a bit more how this form is derived? Im a bit unfamiliar with the (1+(2x+x²))^-1 part came from.

That's a binomial expansion with $2x+x^2$ being the expansion variable (if that's the right term).

 

[lioric]

How did 1/(1+x)^2 Become that?
Could you use the example from the question and show this?

 

[lioric]

How did 1/(1+x)^2 Become that?
Could you use the example from the question and show this?

Ok I figured this out
But, what if you do it in reverse?
1-2x+3x^2-4x^3 and reciprocal that? How does the extra parts get removed from the infinit sequence to for a finite one?
My main problem is that wish to see the cancellation of those

 

[PeroK]

Ok I figured this out
But, what if you do it in reverse?
1-2x+3x^2-4x^3 and reciprocal that? How does the extra parts get removed from the infinit sequence to for a finite one?

If $A = B$ then $B = A$. You can do the binomial expansion in reverse:
$$1 - 2x + 3x^2 - 4x^3 + \dots = (1 + x)^{-2} = \frac 1{(1 + x)^2} = \frac 1 {1 + 2x + x^2}$$

 

[lioric]

If $A = B$ then $B = A$. You can do the binomial expansion in reverse:
$$1 - 2x + 3x^2 - 4x^3 + \dots = (1 + x)^{-2} = \frac 1{(1 + x)^2} = \frac 1 {1 + 2x + x^2}$$

Can’t we reciprocal first and then factorize?

 

[PeroK]

Can’t we reciprocal first and then factorize?

You mean something like this?$$\frac 1 {1 - 2x + 3x^2 - 4x^3 + \dots} = \frac1 {(1 + x)^{-2}} = (1 + x)^2 = 1 + 2x + x^2$$

 

[lioric]

You mean something like this?$$\frac 1 {1 - 2x + 3x^2 - 4x^3 + \dots} = \frac1 {(1 + x)^{-2}} = (1 + x)^2 = 1 + 2x + x^2$$

Yeah but wouldn’t the powers become negetive when you flip them

 

[PeroK]

Yeah but wouldn’t the powers become negetive when you flip them

Not in the world of mathematics I inhabit!

 

[lioric]

Not in the world of mathematics I inhabit!

Could we do it this way ?
1/1^-1-1/(2x)^-1+1/(3x²)^-1

 

[lioric]

Could we do it this way ?
1/1^-1-1/(2x)^-1+1/(3x²)^-1

I’m sorry I’m it didn’t make sense, but this is what I’m trying to figure out

 

[PeroK]

I’m sorry I’m it didn’t make sense, but this is what I’m trying to figure out

I can't help you with that.

 

[lioric]

I can't help you with that.

Ok. Thank you for all that you did. It did help.

 

[lioric]

I can't help you with that.

In conclusion what you are saying is that we can reciprocal the entire thing, but not separately?

 

[PeroK]

In conclusion what you are saying is that we can reciprocal the entire thing, but not separately?

This is elementary:
$$\frac 1{1 + x} \ne \frac 1 1 + \frac 1 x$$

 

[lioric]

This is elementary:
$$\frac 1{1 + x} \ne \frac 1 1 + \frac 1 x$$

Ok thank you