When does this series: \sum_{n=0}^\infty\sin(n!\pi x) converge?

tumahimikgjr

tumahimikgjr

Answered question

2022-02-22

When does this series:
n=0sin(n!πx)
converge?

Answer & Explanation

Abbey Hope

Abbey Hope

Beginner2022-02-23Added 7 answers

I think I can prove it for the first one, and then the others should be similar. In the case of
n=0sin(n!πe)
we need to consider the value of n!πe  mod  π, and it's enough to consider the value of n!e  mod. Now
n!e=n!k=01k!
so that
n!e  mod 1=n!k=n+11k!<k=11(n+1)k=1n0 as n

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