1. ## 2 quick problems

Given that $\displaystyle 1-x+x^2...+ (-x)^n$ is a powers series representation for $\displaystyle \frac{1}{x+1}$, find a power representaion for $\displaystyle \frac{x^3}{1+x^2}$

I can't find out how to do these.

all help appreciated.

2. Why not try to set $\displaystyle x^{2}$ instead of $\displaystyle x$ in series expansion of $\displaystyle \frac{1}{1+x}$ and then multiply each term of the series by $\displaystyle x^{3}$ ?...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. Knowing the sum of an infinite convergent geometric series to be $\displaystyle S_n = \frac{a}{1-r}$ where a is the first term and r is the common ratio, compare this to what they have given you.

For $\displaystyle \frac{x^3}{1+x^2}$ a is equal to x^3 and r is equal to -x^2. The first term will be x^3 and then multiply by -x^2 to generate the rest of the terms.

The power series representation of $\displaystyle \frac{x^3}{1+x^2}$ will be $\displaystyle x^3 - x^5 + x^7 - ... + (-1)^{n}x^{3+2n}$ provided that for the first term n=0

Nevermind, looks like chisigma beat me to it.

4. ok makes sense..

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