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Dropping a higher order term[edit source]

In my proof that the complicated formula for nλ reduces to s sinθ I chose the big-O notation over the Taylor expansion. Halfway through I realized that I was going to unnecessary high order because S=R+O-ε to some power. That lack of forethought caused me to include terms which would only confuse the reader. Here I place what I think is the extra terms that were deleted from the essay:

Copied from deleted version of resource page[edit source]

In order to ensure that the first order calculation is sufficient in the MyOpenMath version of this question, the software verifies that the first order solution is within 10% of the true answer. It is convenient to define so that:

Note from the figure that , and that the two paths are effectively parallel when . The exact formula for the path difference is:


Apparently, if the two paths are nearly parallel, we should be able to show that:


To verify this we can perform a Taylor series of or equivalently use this expansion for small :

The first four terms on the RHS refer to the zeroth, first, second, and third order terms, respectively. The last (fifth-order) term will be briefly discussed but not calculated.

Replacing by into the aforementioned expression, we see that the zeroth, second, and fourth order terms cancel when we subtract:

I replaced by because it is obvious that the fourth order terms also cancel due to the subtraction. In order to obtain a useful value for our small parameter , we divide our expression for by :

where we have defined the small parameter,