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infint series

by fahed faris
(kirkuk\iraq)











































The prime number form a sequence {pn}={2,3,5,7,11,13,…} . It is known that
〖lim〗┬(n--∞)⁡〖(n ln⁡〖n)/pn=1〗 〗 .Using this fact,
Show that ∑_(n=1)^∞▒〖1/pn=1/2+1/3+1/5+1/7+1/11+⋯+1/pn+⋯〗 diverges .

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Dec 20, 2010
Harmonic Series
by: Staff

The question:

The prime number form a sequence {pn}={2,3,5,7,11,13,…} . It is known that
〖lim〗┬(n--∞)⁡〖(n ln⁡〖n)/pn=1〗 〗 .Using this fact,
Show that ∑_(n=1)^∞▒〖1/pn=1/2+1/3+1/5+1/7+1/11+⋯+1/pn+⋯〗 diverges .



The answer:

A harmonic series has the form: Σ(1/n)

The reciprocal prime number series
Σ = ½ + 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + … = ∞
is a harmonic series.

Harmonic series are divergent.

(Only an alternating harmonic series is convergent.)

The proof that this series is a divergent series was demonstrated by the Swiss mathematician Leonhard Euler (1707 – 1783).

A brief discussion of his proof can be found here, under Prop P (Prime Sequence):

http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/harmonic-series

More detailed proofs can be found here:

http://en.wikipedia.org/wiki/Proof_that_the_sum_of_the_reciprocals_of_the_primes_diverges



Thanks for writing.


Staff
www.solving-math-problems.com



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