how to prove this equality ?

[claudiocamera]

I am trying taylor series and Rieman concepts to prove this, but something is missing, can you help me ?

∫ ( y^k) ( e^(-y)) dy = k! --- the limits of the integral are (o, +∞)

 

[matchasm]

How about rewriting to a Gamma function and using one of these:

https://functions.wolfram.com/GammaBetaErf/Gamma2/06/01/01/

The expansion with (n-1)! looks very promising.

 

[vestan]

You must make an Integration by parts Int(udv)=Int(vdu)-[uv]

This integration must make iteratively.

You can see if u=y^k and dv=e^(-y)dy

if you make k iterations you will have k!