Can a convergent sum using only integers produce a complex result? We use this function to define the boundaries for the product in the denominator:
$$f(\text{n$\_$})\text{:=}\frac{1}{8} \left(2 n (n+2)-(-1)^n+1\right)$$  
We calculate the infinite sum:
$$\sum _{n=1}^{\infty } \frac{1}{(f(n)+1)_{f(n+1)-f(n)}}$$ 
We get this complex number:
$$0.61944\, -\text{5.565802539025895$\grave{ }$*${}^{\wedge}$-19} i$$  
Is this a reasonable result?
Note: The calculation takes a very long time in Mathematica, so if this is not reasonable, it's time to perform some debugging.

no, it's not reasonable to get a complex number, when the members of sum are real. We can conclude this, by discussing imaginary parts of both sides.
