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

$\displaystyle (x^{3}+x^4+x^5+...)$

coefficient next to $\displaystyle x^{13}=?$

or a different one:

$\displaystyle (2+3x)^3 \sqrt{2+x}$

coefficient next to $\displaystyle x^5=?$

Could you explain to me how it is done?

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

Quote:

Originally Posted by

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

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

or a different one:

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

You must define the terms and the notation.

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

What does the $\displaystyle ||$ signify?

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 $\displaystyle (x^3+x^4+...)^3$. Is it all right now?

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

Quote:

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 $\displaystyle (x^3+x^4+...)^3$. Is it all right now?

You need to find text material of *generating functions*.

There is a good free pdf generatingfunctionology on the web.

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.