I assume that by sequence they mean
I do not understand what should be ... Is it complex? Usual binomial coefficients have both and integers and . In this problem can be more than ??
Notice that they have "redefined" the numerator of the binomial coefficient as rather than because factorial, is only defined for positive integers. Because they do use the notation "n!" in the denominator, n must be a non-negative integer.
You cannot even ask the question "can n be bigger than " because the complex number system is not an ordered field!
However, "sequence" does NOT mean a sum. That is a "series". They mean just the numbers , , ... themselves. Basically, the first thing you are asked to prove is that there is some real number M such that for all n.