Well, this could easily be a definition
But just using Taylor series for log, we get:
and by continuity of the exponential function, this is equal to
There's actually many ways to define e.
The one I like most is , which comes natural historically (from medieval banking, that is )
Another relatively easy way is by the (all familiar) series , but this is troublesome when it comes to deriving the basic properties of .
The most delicate way, is by describing the function to be the unique solution of the functional equation to satisfy . Then .
Ok, I know you are bored by now, so I stop here ()
But they are not as easy to use for analytical arguments as the one. If you were writing an analysis book I am sure you will use approach because it is by far the simplest to derive all the identities and properties for .
What about ?The most delicate way, is by describing the function to be the unique solution of the functional equation to satisfy . Then .
I' ll disagree here, the lim(1+1/n)^n works fine to that extent.But they are not as easy to use for analytical arguments as the...
You just verified that there were three conditions altogether in that definiton, I just thought hard but couldn't remember if there was another one when making that postWhat about... ?
Here's a proof which involves integrals (But it's just a little integral.)
Let
Since the logarithm is continuous on its domain, we can interchange the function and taking limits.
Make the substitution
Since
So
Take the limit when then by the Squeeze Theorem we can conclude that
Finally