How do I evaluate this sum :∞∑n=1 tan(1n!)arctan(n!) if it is convergent ?.
Note: I think the limit of it's general term is 0 as shown here in WA.
and i will surprised if it is convergent
Note: I edited the question beacuse i meant arctan(n!) in the denominator
Answer
One has, as n→∞,
tan1n!arctan(n!)=tan1n!π2−arctan1n!∼2π⋅1n! then by the comparison test the series ∑n≥1tan1n!arctann! is convergent.
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