I'm a calculus student and I think I have found a new formula to express the gamma function - $\Gamma(x)$ as a limit of an infinite sum. I haven't been able to find such formulas and if somebody knows such formulas please tell me. My formula is that the gamma function of $x$ is equal to
$$\lim_{N\to\infty} \sum_{k=0}^{\infty} \frac{(-1)^kN^{x+k}}{k!(x+k)} \; $$ I have already showed this to my math teacher and she told me she hasn't seen this formula. You can plot it in wolfram alpha for some values and it will match with the gamma function
Yes, your formula is known. It corresponds to the limit $x\to\infty$ of the formula $$ \gamma(s,x)=\sum_{k=0}^{\infty}\frac{(-1)^k x^{s+k}}{k!(s+k)}, \tag{1} $$ where $$ \gamma(s,x):=\int_0^{x}e^{-t}t^{-s-1}dt \tag{2} $$ is the lower incomplete gamma function. Notice that $\lim_{x\to\infty}\gamma(s,x)=\Gamma(s)$.