HI,
What is the power of the set of all sebsets of N on which the series ∑1/n converges to a limit smaller than 1.
Thank's in advance.
For example,the set N in P(N) is not a member in our set (i.e the set that we want to find its cardinality),because the harmonic series diverges.The set of prime natural numbers may belongs because the series based on the prime indeces converges to a limit ,but i don't know if the limit is smaller than 1.
see in google: "harmonic series and its parts".
That still is nonsense. Consider the subset of primes $K=\{2,3,5,7,11,13\}$
$\sum\limits_{x \in K} {{x^{ - 1}}} = {\text{1}}{\text{.40285}}$ see HERE
That finite sum is more than one so it won't do.
Any finite subset of at least two terms one of which is one will not do. Why?
Will the set $T=\{3^{-n}: n\in\mathbb{Z}^+\}$ do? Why or why not?