Inverse of $(1-t)^s$ in the power series ring

I am trying to find the inverse of $(1-t)^s$ in $k[[t]]$. I found out in literature that it should be $\sum\limits_{n \geq 0} \binom{s+n-1}{s-1}t^n$, but I don't see why.

If I expand the element using the binomial theorem, I would have to show that $\sum\limits_{i+j = n} \binom{i+s-1}{i-1}\binom{s}{s-j} = 0$ for $n \geq 0$, but this seems nasty; At least writing out the terms didn't help. Is there a neat proof of this, or maybe a good reference? Thanks!

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1 Answer

Hint $\$ Repeatedly differentiate the series for $(1-t)^{-1}$

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 Wow this is really elegant, thank you. I wonder how I could get through my first undergrad year and not see this trick (my question arose from dealing with hilbert-poincaré series) – jonny Mar 19 '12 at 18:51 I learn every time I visit this site – Siddhi V Iyer Mar 19 '12 at 19:40