sum of 1/A007504(n)

[Max Alekseyev]

On 5/15/07, Stefan Steinerberger <stefan.steinerberger at gmail.com> wrote:

> 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
[...]
> Sum[1/f_n, {i, 1, oo}] < Sum[1/f_n, {i, 1, 10^4}] + 0.000099995 ~ 1.02345802687

Stefan,

As it was pointed out by Fabio Mercurio is a personal email, your
upper bound 1.02345802687 is inconsistent with the value for the sum
of 1/f_n for n up to N=pi(10^10)=455052511:
Sum[1/f_n, {i, 1, N}] = 1.023476323724390... (as I announced earlier)

I have re-done numerical computations following your approach (also
for n up to 10^4) and got the upper bound 1.0235580218780501645...
which is greater than Sum[1/f_n, {i, 1, N}] as expected.

Regards,
Max