sum of 1/A007504(n)

[Stefan Steinerberger]

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