# Thread: Calculate the coefficient of a given term in a power series

1. ## Calculate the coefficient of a given term in a power series

Hello.

I don't know how to calculate given coefficients in a given series. For example

$(x^{3}+x^4+x^5+...)$
coefficient next to $x^{13}=?$

or a different one:

$(2+3x)^3 \sqrt{2+x}$
coefficient next to $x^5=?$

Could you explain to me how it is done?

2. ## Re: Calculate the coefficient of a given term in a power series

Originally Posted by wilhelm
I don't know how to calculate given coefficients in a given series. For example
$(x^{3}+x^4+x^5+...)||$ coefficient next to $x^{13}=?$

or a different one:
$(2+3x)^3 \sqrt{2+x}||$ coefficient next to $x^5=?$

You must define the terms and the notation.

I dare say that most of us have never seen $(x^{3}+x^4+x^5+...)||$ before.

What does the $||$ signify?

3. ## Re: Calculate the coefficient of a given term in a power series

I'm sorry, it doesn't signify anything. I don't know why I put it there. It is simply a series $(x^3+x^4+...)^3$. Is it all right now?

4. ## Re: Calculate the coefficient of a given term in a power series

Originally Posted by wilhelm
I'm sorry, it doesn't signify anything. I don't know why I put it there. It is simply a series $(x^3+x^4+...)^3$. Is it all right now?
You need to find text material of generating functions.

There is a good free pdf generatingfunctionology on the web.

5. ## Re: Calculate the coefficient of a given term in a power series

(x^3 + x^4 + ...) = x^9(1+x+x^2+...) = x^9/(1-x)^3. You can expand the denominator and use a Taylor series expansion and collect the coefficients.
You can try looking up Maclaurin series or Taylor series expansion.