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Binomial expansion

Hey, I would be grateful for any help with this question:

By expanding the following using the binomial expansion, find the first three terms and the general terms in the expansion of f(x) in ascending powers of x, and give the expression for the coefficient of x^n:

f(x) = 1/(1+x) - 2/(x+1)^2

I rewrote f(x) = (1+x)^-1 -2(1+x)^-2

Expanded to give:
1 - x + x^2 ... + 2 - 4x + 6x^2 ...
= -1 - 5x - 5x^2

Firstly, is this expansion correct? And then I'd really appreciate any guidance how to find the expression for the coefficient of x^n.

Thank you :smile:
Original post by sqrt123
Hey, I would be grateful for any help with this question:

By expanding the following using the binomial expansion, find the first three terms and the general terms in the expansion of f(x) in ascending powers of x, and give the expression for the coefficient of x^n:

f(x) = 1/(1+x) - 2/(x+1)^2

I rewrote f(x) = (1+x)^-1 -2(1+x)^-2

Expanded to give:
1 - x + x^2 ... + 2 - 4x + 6x^2 ...
= -1 - 5x - 5x^2

Firstly, is this expansion correct? And then I'd really appreciate any guidance how to find the expression for the coefficient of x^n.

Thank you :smile:


Not quite. You have the wrong signs on each term of the second expansion.

For a general coefficient of xnx^n, you should determine what it is first of all in (1+x)−1(1+x)^{-1} and then similarly what it is in −2(1+x)−2-2(1+x)^{-2} then add them together.
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sqrt123
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Original post by RDKGames
Not quite. You have the wrong signs on each term of the second expansion.

For a general coefficient of xnx^n, you should determine what it is first of all in (1+x)−1(1+x)^{-1} and then similarly what it is in −2(1+x)−2-2(1+x)^{-2} then add them together.

Okay so it's -1 +3x -5x^2 ...
and for the first part I think the coefficient is just given by (-1)^n, and the second part is 2(n+1)(-1)^n

Thanks for your help !
(edited 4 years ago)
Original post by sqrt123
Okay so it's -1 +3x -5x^2 ...
and for the first part I think the coefficient is just given by (-1)^n, and the second part is 2(n+1)(-1)^n

Thanks for your help !


That's right for the first one.

But for the second one, it's slightly off. And you can check this yourself. When n=0 you're saying the coeff of x0x^0, i.e. the constant term, is 2. When it's actually -2.

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