Question: Why does the summation not work?

I have a say sum(F(k) ,k=1..n)  It fails unless n is an actual integer.  But sum(F(k), k=1..a) works. Then then eval(%,a=n) completes it. That is probably correct but not necessarily a valid assumption. Details in attached. Also it is hard to understand the error message.
 

restart

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Sinta := (k^2+n^2-k)*n/((k^2+n^2-2*k+1)*(k^2+n^2))

(k^2+n^2-k)*n/((k^2+n^2-2*k+1)*(k^2+n^2))

(1)

Ai := sum(Sinta, k = 1 .. n)

-((1/2)*I)*(2*n^2+I*n)*Psi(n-I*n)/(4*n^2+1)+((1/2)*I)*(2*n^2-I*n)*Psi(n+I*n)/(4*n^2+1)+((1/2)*I)*(-2*n^2+I*n)*Psi(n+1-I*n)/(4*n^2+1)-((1/2)*I)*(-2*n^2-I*n)*Psi(n+1+I*n)/(4*n^2+1)+((1/2)*I)*(2*n^2+I*n)*Psi(-I*n)/(4*n^2+1)-((1/2)*I)*(2*n^2-I*n)*Psi(I*n)/(4*n^2+1)-((1/2)*I)*(-2*n^2+I*n)*Psi(1-I*n)/(4*n^2+1)+((1/2)*I)*(-2*n^2-I*n)*Psi(1+I*n)/(4*n^2+1)

(2)

Souta := n/(k^2+n^2-k)

n/(k^2+n^2-k)

(3)

NULL

sum(Souta, k = 1 .. n)

Error, (in assuming) when calling 'Dfnt_4'. Received: 'when calling 'Dfnt_4'. Received: 'when calling 'unknown'. Received: 'invalid input: Dfnt_4 expects its 3rd argument, fpts, to be of type Or(list, piecewise), but received 0'''

 

"(->)"

Error, invalid input: evalf[10] expects 1 argument, but received 0

 

 

 

This works

sum(Souta, k = 1 .. a)

n*Psi(a+1/2-(1/2)*(-4*n^2+1)^(1/2))/(-4*n^2+1)^(1/2)-n*Psi(a+1/2+(1/2)*(-4*n^2+1)^(1/2))/(-4*n^2+1)^(1/2)-n*Psi(1/2-(1/2)*(-4*n^2+1)^(1/2))/(-4*n^2+1)^(1/2)+n*Psi(1/2+(1/2)*(-4*n^2+1)^(1/2))/(-4*n^2+1)^(1/2)

(4)

Ao := eval(%, a = n)

n*Psi(n+1/2-(1/2)*(-4*n^2+1)^(1/2))/(-4*n^2+1)^(1/2)-n*Psi(n+1/2+(1/2)*(-4*n^2+1)^(1/2))/(-4*n^2+1)^(1/2)-n*Psi(1/2-(1/2)*(-4*n^2+1)^(1/2))/(-4*n^2+1)^(1/2)+n*Psi(1/2+(1/2)*(-4*n^2+1)^(1/2))/(-4*n^2+1)^(1/2)

(5)

"(->)"

n*Psi(n+.5000000000-.5000000000*(-4.*n^2+1.)^(1/2))/(-4.*n^2+1.)^(1/2)-1.*n*Psi(n+.5000000000+.5000000000*(-4.*n^2+1.)^(1/2))/(-4.*n^2+1.)^(1/2)-1.*n*Psi(.5000000000-.5000000000*(-4.*n^2+1.)^(1/2))/(-4.*n^2+1.)^(1/2)+n*Psi(.5000000000+.5000000000*(-4.*n^2+1.)^(1/2))/(-4.*n^2+1.)^(1/2)

(6)

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