MATLAB: Symsum equation not matching

symsum equation mismatch incorrect

I am solving an equation to get the result in terms of symbol 'SI'. This is the given equation
but answer in terms of SI i am getting is not correct. I have used following code:
A=symsum(   2*m/( SI*SI*(NI-SI+1 )),m,1,SI/2   ) +  symsum( 1/ ( SI*SI*(NI-SI+1 )),m,SI/2+1,SI )
which becomes
A = 1/(2*SI*(NI - SI + 1)) + (SI + 2)/(4*SI*(NI - SI + 1))
solving for SI with A=0 gives result as
ans =
 root(z^3 - (z^2*(2*NI + 2))/2 - z/2 - 1, z, 1)
 root(z^3 - (z^2*(2*NI + 2))/2 - z/2 - 1, z, 2)
 root(z^3 - (z^2*(2*NI + 2))/2 - z/2 - 1, z, 3)
when i use vpa(ans), i get 3 equations with long terms (without z) but still answers does not match the given result shown below in diagram (NI is substituted by SL/u in the document so in square root term is actually (2/3)*NI-11/36
Any help will be highly appreciated.

Best Answer

No, the SI you give is not a solution to that summation being 0.
2/(SI^2*(NI-SI+1)) is independent of m, so the first term of A is
 2/(SI^2*(NI-SI+1)) * Sum(m, m=1..SI/2)
Sum(m, m=1..SI/2) for positive SI is ((SI/2)+1)*(SI/2)/2 . So you can calculate the first term as 2/(SI^2*(NI-SI+1)) * ((SI/2)+1)*(SI/2)/2 which works out as (SI+2)/(4*SI*(NI-SI+1))
1/(SI*(NI-SI+1)) is independent of m, so the second term of A is
1/(SI*(NI-SI+1)) * Sum(1, m=SI/2+1..SI)
which is 1/(SI*(NI-SI+1)) * (SI - (SI/2+1) + 1) = 1/(SI*(NI-SI+1)) * SI/2 which works out as 1/(2*(NI-SI+1))
so A = (SI+2)/(4*SI*(NI-SI+1)) + 1/(2*(NI-SI+1)) which works out as
 A = (3*SI+2)/(4*SI*(NI-SI+1))
Note: this assumes that SI/2 is a positive integer! If it is not then the summation bounds need to be more carefully defined!
Now, for (3*SI+2)/(4*SI*(NI-SI+1)) to be 0, the numerator must be 0, which requires that 3*SI+2 is 0, which would require that SI = -2/3.
SI = -2/3 violates the assumption that SI/2 is a positive integer.
Therefore there is no solution for A = 0 -- not unless you very carefully define what it means to sum with non-integer boundaries or what it means to sum from 1 to a negative number in the bounds.
Exception: if NI is infinite but SI is finite then A can be 0.
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