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#1 2018-04-12 22:59:10

zahlenspieler
Member
Registered: 2018-04-12
Posts: 3

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)

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