Which are the "First Two Terms" of a Binomial Expansion?

Question

When asked to find the first two terms of a binomial expansion (or, a power series in general), does 1 (i.e. the $x^0$ term) count as the first term?

Answer 1

A binomial expansion is sorted by degree, and if 1 (degree 0) is in the first two terms, then yes.

Answer 2

Yes, but it's slightly ambiguous.

In general, "the first term" always means the leftmost or initial term unless otherwise specified. But sometimes, "the $i$th term" can refer to the term with index $i$. In cases where which of these two conventions is being used is not clear from the context, the writer should specify.

Answer 3

It depends on what you are expanding, and what you consider 'first'.

A typical list order is the descending power series of the leftmost terms in the factor.

• $(x+y)^n$ has 'first' two terms of $x^ny^0$ and $n x^{n-1}y$.
• $(1+x)^n$ has 'first' two terms of ...
• $(x^2-2xy+y^2)^n$ has 'first' two terms of ... ... ...

However, this is not always the case.