Math Is Fun Forum

zahlenspieler

Member

Registered: 2018-04-12

Posts: 7

Towards a proof of Euler's formula

Hi everyone!
It all started when I proved

.
Let

,
where r is some real number.
Now I would like to show that

,
where

.
So I proved the following inequality for positive integers

:

.

Expanding

gives

.
As n grows towards infinity, the binomial coefficients go to

. Now taking the absolute values, that sequence is bounded above by

.

But now I got stuck. One idea that I have is the sandwich theorem -- i.e. to 'squeeze' the sequence of partial sums of the cosine series between that sequence above and another sequence

.

The trouble is the altering signs, so I just can't add up inequalities ...
Any ideas?
Regards,
zahlenspieler

Last edited by zahlenspieler (2018-04-19 17:10:29)