Re: convergence of series involving arctan, arcsin or arccos.

In article <170120061302463137%bruck@xxxxxxxxxxxxxxxxx>,
Ronald Bruck <bruck@xxxxxxxxxxxxxxxxx> wrote:
>In article <1137522357.911833.126260@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
>katy <mcld@xxxxxxxxxxxxxxx> wrote:

>> I'm having some problems in finding if a series is convergent or
>> divergent when the geral term involves arctan, arcsin or arccos.
>>
>> For example,
>>
>> a series from n=1 to infinity with term: (3^n arcsin((-1)^n/n) )/n!
>>
>> Did i have to see the limit of what is inside the arcsin?
>
>The arcsin term is "gingerbread". Look at the rest of the product:
>3^n/n!. Can you prove that converges? (It does, quite rapidly.)
>
>Now you also have the complicated factor arcsin ((-1)^n/n). But I
>called it "gingerbread"; you could just as well have
>
> arcsin( (-1)^n arctan(n * sin(n)) )
>
>or anything else inside there. All that matters is that arcsin is
>bounded. All the rest is, well... "gingerbread".

On the other hand, if you had something like
sum_{n=1}^infty arcsin((-1)^n/n)
or
sum_{n=1}^infty arcsin(1/n)/n
or (a tougher one)
sum_{n=1}^infty (arcsin(1 - 1/n) - pi/2 + sqrt(2/n))
you would need to know more of the properties of the arcsin function.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

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