<s>[INST] General formula for the power series of $\dfrac{1}{(1+x)^3}$ I wish to find the general formula for the following power series:
$\dfrac{1}{(1+x)^3}=1-3x+6x^2-10x^3+15x^4-21x^5+28x^6...$
The difference between the first and second term is $2$
The difference between the second and third term is $3$
The difference between the third and fourth term is $4$
The difference between the fourth and fifth term is $5$
and so on.
How do you find the general formula for this series? It is not quite an arithmetic series, and definitely not a geometric series, so what should I do?
 [/INST] If you know calculus, then
$\frac {1}{(1+x)^3} = \frac {d^2}{dx^2}\frac {1}{2(1+x)} = \frac {d^2}{dx^2} \sum_\limits{n=0}^\infty \frac {(-1)^nx^n}{2} = \sum_\limits{n=0}^\infty \frac {(-1)^n(n+1)(n+2) x^n}{2}$
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