sum of 1/A007504(n)
Maximilian Hasler
maximilian.hasler at gmail.com
Wed May 16 01:20:15 CEST 2007
Modulo exchanging some i <-> n I agree with your proof and find it
more convincing than the numerical computations (including mine of
course) in view of the slow convergence (clearly, Simon's ...0544..
will soon become ...0545... and sooner or later, ...0550.... etc...)
On 5/15/07, Stefan Steinerberger <stefan.steinerberger at gmail.com> wrote:
> Hello,
>
> how about the following argument?
>
> Let f_n be the sum of the first n primes. Then it is easy to see that
> f_n = 2 + 3 + 5 ... + p(n) > 1 + 3 + 5 + ... + (2n-1) = n^2
>
> Now we use this argument to get an upper bound for Sum[1/f_n, {i, 10^4 + 1, oo}]
>
> Sum[1/f_n, {i, 10^4 + 1, oo}] < Sum[1/n^2, {i, 10^4 + 1, oo}]
>
> Now we can easily evaluate the last sum, for
>
> Sum[1/n^2, {i, 10^4 + 1, oo}] = pi^2/6 - Sum[1/n^2, {i, 1, 10^4}]
>
> Mathematica gives the approximate value of 0.000099995 (perhaps someone
> can confirm that?). We can combine those facts to obtain
>
> Sum[1/f_n, {i, 1, oo}] < Sum[1/f_n, {i, 1, 10^4}] + 0.000099995 ~ 1.02345802687
>
> Since pi/6 + 1/2 is bigger than this upper bound, we can conclude that
> the sum indeed does not converge against the proposed limit.
>
> Any mistakes?
>
> Stefan
>
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