I am trying to find the coefficient of $x^{n}$ where $n \in[0, \infty)$ in the expansion
$(x^0 + x^1 + x^2 + x^3 + x^4 + x^5 … +x^{\infty}$)$^{t}$, where $t$ is an integer.
I haven't been able to distinguish a pattern between the coefficients and the value of $t$, but the sum $\sum_{i = 0}^{\infty} x^i$ is the Taylor series expansion of the geometric sum formula. I also tried some things with the binomial theorem, but was not able to make too much progress.
Best Answer
Assuming $|x| < 1$ use GP sum and the expression reduces to,
$$(1-x)^{-t}$$
Coefficient of $x^n$ in $(1-x)^{-t}$ equals $\binom{n+t-1}{t-1} = \binom{n+t-1}{n}$.
Check out the binomial expansion of $(1+x)^m$ when $m$ is negative.