Let $p_1=2
Let Sn=∞∑k1=0∞∑k2=0…∞∑kn=01pk11pk22…pknn
It is easy to check that S1=2,S3=3 and in general Sn=11−1p111−1p2…11−1pn
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 Sn∼f(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=∞∑k1≠k2≠⋯≠kn1pk11pk22…pknn
Here k1≠k2≠⋯≠kn means they are distinct(n! values).
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