I'm trying to figure out the theory behind how to obtain an equation, the whole thing is very long so I couldn't post the entire thing here, but part of it requires a Taylor expansion in certain terms and I've completely forgotten how to do so.
The Taylor expansion is of $\frac{(u + v)}2$ and $\frac{(u - v)}2$ where $$u = \frac{2{\pi}x}{\lambda_{Na}+\frac{\Delta\lambda}2}$$ and $$v = \frac{2{\pi}x}{\lambda_{Na}-\frac{\Delta\lambda}2}$$ but each expansion is supposed to be in powers of $\Delta\lambda$ since $\Delta\lambda << \lambda_{Na}$. Also, I'm told to only retain one term (the first non-vanishing term) in each series. I'm not sure what to do for this and am at a loss. Help would be very greatly appreciated.
Let me know if there's anything I left out that could be useful and I'll see what I can do.