See my 2 posts here.
It is not a secret for anyone that I'm a 'fan' of Leonhard Euler . One of his most beautiful 'discoveries' is the complex of properties of the constant that brings his name...
(1)
The first problem concening Euler's constant I will propose to You is : starting from definition (1) demonstrate that is...
(2)
Kind regards
Congratulations for Paul and SP!...
A similar but more 'general purpose' way to arrive at the result is the use of the following 'nice lilttle formula'. From the well known definition of [negative] exponential function...
(1)
... derives 'spontaneously' the formula...
(2)
... that with the substitution becomes...
(3)
... and with the substitution becomes...
(4)
Now if in (4) we set we obtain...
(5)
If now we remember the identity...
(6)
... we arrivwe at the final result...
(7)
Kind regards
The correct procedure requires to remember that by definition is...
(1)
... so that the limit...
(2)
... exists if the limit (1) exists. In your example is...
(3)
... and none of the exists so that the expression...
(4)
... is a nonsense. Anyway your observation is probably correct so that I propose to overcame any difficulty with the following formal statement...
(5)
Kind regards
This is in not really more self evident than the previous assertion. However, simplependulum provides us with a proof. Indeed,
(this also holds for non-integer values of as soon as they're greater than 1, cf. the proof)
Hence, from this inequality, for ,
.
(Provided the integral is finite, so under mild hypotheses on )
I copied this proof from a book which talks about elements of complex analysis , almost the last chapter .
Perhaps this excellent proof is completed by some GREAT Mathematicians , if i have enough time , i will complete the proof here . We can see that some parts of the original proof are missing ( e.g. proof of ) .
Hence, from this inequality, for ,
.
(Provided the integral is finite, so under mild hypotheses on )
This inequality appears at the end of the proof in my book ! This is the main point , when n tends to infinity , their difference ( ) tends to zero .
Great !