I am asking my mentor for my class about this question, but unfortunately the answers I get from her often take forever and do not always clear things up for me, so I hope someone out there in Physics Forum has a good way of explaining this to me. Here goes...(adsbygoogle = window.adsbygoogle || []).push({});

The question has some Mathematica input (Normal[Series] command) and output for the function f(x) = e^{-x}Cos(x). Then the question is asked "Mathematica is trying to tell you that, at x = 0, e^{-x}Cos[x] and 1 - x + (x^{3})/3 - (x^{4})/6 + (x^{5})/5 have order of contact ____?"

Now to answer, I went back to what we learned about the definition of an expansion in powers of x was. This definition from the materials is: Given a function f[x], the expansion of f[x] in powers of x is: a[0] + a[1]x + a2[x]^{2}+ ... + a[k]x^{k}, where the numbers a[0], a[1], a[2], ..., a[k] are chosen such that for every positive integer m, the function f[x] and the polynomial a[0] + a[1]x + a[2]x^{2}+ ... + a[m]x^{m}have order of contact m at x = 0.

From this definition, I believed the answer to the question would be "5" because we see that in the expansion given, we have a 5th degree polynomial. Instead, the answer was 4. Can you please explain where my reasoning was faulty, and why the actual answer is 4? My guess is that the (x^{5})/5 term is actually x^{m+1}making it a 5th degree polynomial where m = 4. Is that correct? Or am I barking up the wrong tree?

Thank you for your help!

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# Expansions and order of contact

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