sum of 1/A007504(n)

Stefan Steinerberger stefan.steinerberger at
Wed May 16 22:40:30 CEST 2007

> 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.

Thanks for double-checking, I wasn't too sure about the numbers anyway.
As you have already indicated, the approach can be generalized and
gives the following upper bound for all natural numbers n.

Sum[1/f_i, {i,1,Infinity}] < Sum[1/f_i, {i,1,n}] + pi^2/6 - Sum[1/i^2, {i,1,n}]

T.D. Noe has already remarked that the sum over the first n primes
grows approximately like 0.5*n^2*log(n). Perhaps someone can improve
the lower bound (currently n^2). An improved lower bound would lead
to better upper bounds - although they might involve infinite series who
cannot be easily calculated, which might nullify the advantage gained
by the accelerated convergence.


Straightforward duplicates:

A098622 and A099698
A070832 and A094212
A056255 and A101842
A068121 and A077735
A048603 and A048609
A120722 and A120723
A056251 and A101825
A093469 and A093816
A010928 and A080559

Possible duplicates:

A096053 and A107905 (after adjusting for offset in A107905)|id:A107905

A096299 and A110382 (possibility noted in comments for A110382)|id:A110382

A085903 and A107664 (the G.F. for A085903 has an extra "x" in the numerator,|id:A107664

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