suppose that {Pk} is a set of prime numbers.tell me is the series of 1/Pk is Convergent or divergent?and why?(

Question

for all k>=1,k is the index of prime p!(series is 1/P1+i/P2+1/P3+...)

Answer 1

Suppose the series is 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + . . . + 1/Pn.

t has been known for quite a long time that, although the terms converge to zero, the sum diverges. It is asymptotic to ln(ln(Pn)).

Answer 2

The people who are saying "converges to 0" are mixing up the concept of sequence and series. The sequence 1/Pk converges to 0. The series sum(1/Pk) diverges to positive inifinity as other posters indicated.

Answer 3

if me memory of cal 2 is correct this is a divergent series.
even though 1/Pk>1/P(k+1)...eventually 1/Pk--->0

but each 1/Pk adds something more to the series.
So the series never reaches a finite val.

Answer 4

This series diverges. See the following website,
where you will find five proofs of this:
http://en.wikipedia.org/wiki/Proof_that_the_sum_of_the_reciprocals_of_the_primes_diverges
On the other hand, it is also known that the
sum of the reciprocals of all twin primes is convergent
(or perhaps finite).

Answer 5

It is convergent to 0. Each term in Pk increases to infinity, so 1/Pk approaches o as k increases.

Answer 6

Pk is positively increasing so the denominator in 1/Pk is continuously positively increasing so the set converges to zero :)

Answer 7

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