one pound

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These are questions asked by one pound

In this post https://math.stackexchange.com/questions/1196194/asymptotic-behaviour-of-frac1x-1-frac1x2-1-frac1x3-1-cdots

this sum is evaluated in the answer to be

I cannot see how to use eularmac to get this answer as x approaches 1.

 

Maybe using

ge := `assuming`([sum(1/(x^k-1), k = 1 .. infinity)], [k::integer])then something like

 

eulermac(ge, limit(ge, x = 1), 3)

I cannot seem to set up this integral correctly:

Gradshteyn 3.529

I think maple should do it.

`assuming`([int(((cosh(a*x)-1)/sinh(b*x))(1/x), x = 0 .. infinity)], [b > abs(a)])

I wonder where I'm going wrong here? Thank you in advance.

I am trying to get the exponential Fourier series versiion  with C_n seperate and it appears this package https://www.maplesoft.com/applications/view.aspx?SID=4857&view=html seems to do this

While I'm not particular to this package https://www.maplesoft.com/applications/view.aspx?SID=4857&view=html I cannot get it to work. It could be that FourierSeries is not built into maple in which case only the worksheet appears to be availble on the link above with no application in which case it should be removed.

If not an alternative would do. For example I got this https://www.maplesoft.com/applications/view.aspx?SID=33406 to work but I cannot get it in  format like Khanshan's package see example output (1.7).

Is there any maple function to test whether infinite sums and integrals can be swapped say for example whether

this is true? for a given f(x):

Is there a riemann sum to integral converter?

If not then,

So I thought I could make some progress using the maple limit command in the this problem:

limit(Sum(csc(Pi*x/i)^3*sin(Pi*x)^3/i^3, i = 1 .. n), n = infinity)

which gives signum(sin(Pi*x)^3/x^3)*infinity

but what does this mean? 

How do I convert Sum(csc(Pi*x/i)^3*sin(Pi*x)^3/i^3, i = 1 .. n), n = infinity to an integral or from this result?

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