Question

An attempt to write a series representation for the gamma function

Answer 1

I know that\[\large \Gamma(n)=\int\limits_{0}^{\infty}\frac{x^{n-1}}{e^x}~dx\]

Answer 2

don't help me yet

Answer 3

we're just watching :-)

Answer 4

\[\large e^{-x}=\sum_{n=0}^{\infty}\frac{x^n}{n!}\]\[\large x^{t-1}e^{-x}=\sum_{n=0}^{\infty}\frac{x^{n+t-1}}{n!}\]then integrate both sides,\[\large \Gamma(t)=\int\limits_{0}^{\infty}x^{t-1}e^{-x}~dx=\lim_{C_1 \rightarrow \infty} \left( \sum_{n=0}^{\infty}\frac{x^{n+t}}{(n+t)n!}~{\Huge|}^{C_1}_{0}\right)\]

Answer 5

I forgot something

Answer 6

\[\large \Gamma(t)=\int\limits_{0}^{\infty}x^{t-1}e^{-x}~dx=\lim_{C_1 \rightarrow \infty} \left( \sum_{n=0}^{\infty}\frac{(-1)^nx^{n+t}}{(n+t)n!}~{\Huge|}^{C_1}\right)\]

Answer 7

wrote the (-1)^n that I left out

Answer 8

it looks right

Answer 9

very cool, tnx:)
I thought I did something crazy