How <span title="(1-x)^(-1)=1+x+x^2+x^3+x^4+x^5+x^6+x^7+................?">(1-x)^(-1)=1+x+x^2+x^3+x^...</span>

Question

How (1-x)^(-1)=1+x+x^2+x^3+x^...

Answer 1

I believe you meant to post this question in the Math section !

(I answered some of your other questions there)

There are at least 2 ways to get this:

One way is to simply do the long division:

       ______

1-x | 1          

which gives the infinite series you have in the question.

The other way is to derive formally the Taylors series for this function at x=0:

f(x) = f(0) + f&#039;(0)x/1! + f&#039;&#039;(0)x^2/2! + ........

So you need to calculate :

f(0) = 1

f&#039;(x) = 1/(1-x)^2   f&#039;(0) = 1

f&#039;&#039;(x) = 2/(1-x)^3  f&#039;&#039;(0) = 1

...

The result is 1/(1-x) 1 + x + x^2 + .....

Note again that this is only valid for |x|&lt;1 as mentioned in the other answers I posted.

Answer 2

Que?