Friday, 19 June 2015

sequences and series - Is this :suminftyn=1fractan(frac1n!)arctan(n!) convergent sum?



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!π2arctan1n!2π1n! then by the comparison test the series n1tan1n!arctann! is convergent.


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