[seqfan] Re: towards a new series for e

[Olivier Gerard]

Hello,

I am not sure of what you are searching for, Jaume,
but you can slightly improve your interesting approximation of
e by

e ~= H_8( 1 +  1/(80)^2  + 3 / (20000)^2  )

(beware that you did not give the right sign in your original message)

and you can go on, as this is mainly recoding the decimal
development of  e/H_8.  But I am not sure there is a good
mathematical reason why H_8, which is just a (small) fraction
among others, while e is transcendental,
would be the right starting point or
the right factor for a defining series of e .

Another way of looking at this is to say that e is the

8.00361474264417023520337... th Harmonic Number

while pi is the

12.489356501947330060412... th

and gamma the

0.461632144968362341262659... th

as the notion of Harmonic Number can be made continuous

Perhaps these developments should be in the OEIS.


Olivier


On Mon, Nov 9, 2009 at 19:15, Jaume Oliver i Lafont
<joliverlafont at gmail.com> wrote:
> Hello,
>
> e~=H_8(1-1/80^2) is equation (9) in
> http://mathworld.wolfram.com/eApproximations.html
>
> There might be a simple series whose first two terms yield this approximation,
> but i have failed several times to find it.
>
> Can you do it? At least next term inside the parenthesis?
>
> Jaume
>