so if then
Thank you Moo, that is much simpler.
Here is another proof I thought of today. Taking the logarithmic derivative of , we get , and this goes to as . Hence the function satisfies for all complex , and it follows that
Edit : oops, this is exactly what Random Variable did
We develop .
For a fixed , the ratio tends to 1 (the numerator is a monic polynomial of degree in the variable ).
"Formally", we thus have .
The good point of this proof is that we can "see" why the limit is the exponential; this may have been Euler's way to prove the formula. However, it is not trivial to prove the limit rigorously (there is a summation in , not a fixed index...). This can probably be done elementarily, but here is a short not-so-elementary way. We have where for , and otherwise. For fixed , . The previous ratio is bounded by 1, hence for all . We have , therefore we can apply the bounded convergence theorem to take the limit in in the summation.
For the function to be its own derivative, we would require that
and for simplicity, we will now use the symbol to represent this particular value of .
So we have
Now by letting , we notice that as .
Can you go from here?
Also in (**), you can not take the power outside the limit either since it is not a constant, i.e., depends on the limit variable...
A simple note.
provided that .