Series expansion/algebra problem

[BOAS]

Hi,

i'm revising series expansions and my problem has arisen from an example in my textbook, but it's not directly related to the series expansion itself. It's more of an algebra question, where they've made a rearrangement I can't follow...

Homework Statement

If $n$ is a positive integer find the coefficient of $x^{r}$ in the expansion of $(1 + x)(1 - x)^{n}$ as a series of ascending powers of x.

Homework Equations

The Attempt at a Solution

The solution starts by stating the following expressions are equivalent.

$(1 + x)(1 - x)^{n} \equiv (1 - x)^{n} + x(1 - x)^{n}$

If I take this statement to be true, I can follow through the rest of the example without a hitch, but I just can't see what they've done here.

Thanks for any help you can give,

BOAS.

 

[Saitama]

Hi,

i'm revising series expansions and my problem has arisen from an example in my textbook, but it's not directly related to the series expansion itself. It's more of an algebra question, where they've made a rearrangement I can't follow...

Homework Statement

If $n$ is a positive integer find the coefficient of $x^{r}$ in the expansion of $(1 + x)(1 - x)^{n}$ as a series of ascending powers of x.

Homework Equations

The Attempt at a Solution

The solution starts by stating the following expressions are equivalent.

$(1 + x)(1 - x)^{n} \equiv (1 - x)^{n} + x(1 - x)^{n}$

If I take this statement to be true, I can follow through the rest of the example without a hitch, but I just can't see what they've done here.

Thanks for any help you can give,

BOAS.

Let $(1-x)^n=a$, then $a(1+x)=a+ax$. Do you see now? :)

 

[BOAS]

Let $(1-x)^n=a$, then $a(1+x)=a+ax$. Do you see now? :)

Perfectly.

Thank you!