Monday, 23 June 2014

sequences and series - divergence of suminftyk1neqk2neqdotsneqknfrac1pk11pk22dotspknn

Let $p_1=2


Let Sn=k1=0k2=0kn=01pk11pk22pknn



It is easy to check that S1=2,S3=3 and in general Sn=111p1111p2111pn



Then if we can show that as n, Sn diverges to infinity we can say n=11n also diverges to infinity.



If we can show that Snf(n), where f(n) increases to infinity as n, then we are done.



What is this f(n)?




Is there any other way to show k=11k diverges using Sn?






But I have no idea about the following problem:



Tn=k1k2kn1pk11pk22pknn



Here k1k2kn means they are distinct(n! values).




How should we proceed to in this case?

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