If I factor $(n+1)/2n$ by $n$, I get $n(1+1/n)/n(2)$. Simplifying, I end up with $(1 + 1/n) / 2$.
This can be rewritten as: $1/2 + (1/n)/2$, which would give me $1/2 + (1/n) \times (1/2) = 1/2 + 1/(2n)$.
Why is this wrong? I'm told: $(n+1) / (2n) = 1/2 + 1/n$ ?
If your n+1/2n represents $\frac{n+1}{2n}$, then we have$$\frac{n+1}{2n}=\frac{n}{2n}+\frac{1}{2n}=\frac{1}{2}+\frac{1}{2n}$$ because $$\frac {A+B}{C}=\frac{A}{C}+\frac{B}{C}.$$