Convergence of subseries of the harmonic series

[hnbc1]

I need to show that the by eliminating infinitely many terms of the harmonic series, the remaining subseries can be made to converge to any positive real numbers.

I have no clue to prove this. I know harmonic series diverges really slowly, will this fact come into play?

Thank you very much!

 

[lanedance]

say you want to sum to L, consider something like the following:

if $1 < L$, start sum $S(1) = 1$, otherwise $S = 0$
now if
if $S(1)+ \frac{1}{2} < L$ sum $S(2) = S(1)+\frac{1}{2}$, otherwise $S(2) = S(1)$

and consider carrying on this process...

 

[lanedance]

as you mentioned this is helped by the fact the terms of the harmonic series tend to zero, so as your sum approaches the required value form the left hand side, you can always find terms smaller than the remaining gap

 

[hnbc1]

Thanks, lanedance.
I think the idea is pretty straightforward, but I need more efforts to prove it. I'll figure it out, thank you!

 

[Char. Limit]

If this is true, wouldn't it imply that any positive real number can be written as a sum of reciprocals of certain numbers?

 

[Office_Shredder]

If this is true, wouldn't it imply that any positive real number can be written as a sum of reciprocals of certain numbers?

As an infinite sum of reciprocals yes.

 

[hnbc1]

I think so.