Final answer:
The expansion of arctan(x)/(1-x) is best done with the Maclaurin series or Taylor series, rather than the binomial theorem. The question involves using series expansions like the binomial theorem, but the function's form suggests using Taylor or Maclaurin series for the appropriate expansion.
Step-by-step explanation:
The question is asking to find the expansion of the function arctan(x)/(1-x) using series expansions. Specifically, the student is referred to binomial series, Taylor series, power series, and Maclaurin series. These are different methods used to approximate functions with an infinite series of terms, where each term is a multiple of the function's derivatives evaluated at a particular point.
The binomial theorem can be used when expanding expressions of the form (a + b)n, and it is given by:
(a + b) = an + nan-1b + n(n-1)/2!an-2b2 + n(n-1)(n-2)/3!an-3b3 + ...
However, for the function arctan(x)/(1-x), the Maclaurin series or Taylor series would be more appropriate. The Maclaurin series is a special case of the Taylor series around 0, and both involve expressing a function as an infinite sum of terms calculated from the values of its derivatives at a single point.