How should I expand 1/(1+x)^n around x=0?

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Homework Help Overview

The discussion revolves around expanding the function 1/(1+x)^n around x=0, specifically seeking a polynomial form rather than the reciprocal of a polynomial. The subject area includes calculus and series expansions.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants suggest considering a Taylor expansion and inquire about the original poster's attempts and challenges. There is mention of the binomial theorem for expanding (1+x)^n, but the focus is on the negative exponent and its polynomial expansion.

Discussion Status

The discussion is ongoing, with participants exploring different methods of expansion. Some guidance has been offered regarding the Taylor series, but there is no explicit consensus on the approach to take.

Contextual Notes

There is a repeated emphasis on expanding around x=0 and the need for a polynomial form, which suggests constraints in the original poster's understanding or approach to the problem.

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Homework Statement



How should I expand 1/(1+x)^n around x=0?

Homework Equations


The Attempt at a Solution

 
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Did you try a Taylor expansion?
What did you get? Where did you run into problems?
 
mfb said:
Did you try a Taylor expansion?
What did you get? Where did you run into problems?

I know that (1+x)^n could be expanded easily by binomial theorem, but what I need here is to expand (1+x)^-n into polynomial form, not the reciprocal of a polynomial
 
I don't see how your answer is related to my post.
You can just calculate the Taylor expansion.
 
liyz06 said:

Homework Statement



How should I expand 1/(1+x)^n around x=0?

Homework Equations



The Attempt at a Solution

Are you familiar with Taylor Series ?

The Taylor expansion for a function, f(x), expanded about x = a is:

## \displaystyle f(x)=f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots \ . ##

So, expanding about x = 0 gives:

## \displaystyle f(x)=f(0)+\frac {f'(0)}{1!} (x)+ \frac{f''(0)}{2!} (x)^2+\frac{f^{(3)}(0)}{3!}(x)^3+ \cdots \ . ##
 

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