Exercise 13

Gary

The only problem is that (14) on page 73 is a statement of the Binomial Theorem for any real number. Also it is just a statement, not a proof. In the following paragraph Needham goes on to point out that exercises 16,17 and 18 of chapter 2 show that (14) is true for all negative integer powers and then for all rational powers. This effectively covers everything except complex numbers.

He then writes, in the same paragraph, "Later we shall use other methods to establish a still more general version of (14) in which the power n is allowed to be a complex number!" (Needham's punctuation).

I have always assumed that he was referring to exercise 13 chapter 5 here and so I think it would not be correct to assume (14) is true for complex values of n in the proof of this exercise (13). Although I do not mention it, I am assuming the result of exercise 12 chapter 5 in my published solution to exercise 13. Maybe I should amend my solution to point this out.

The way I see this exercise is that since a is defined to be an arbitrary constant, it is a demonstration that the Binomial Theorem (14) on page 73 is valid for any n and in particular for complex values of n.

What do you think?

Vasco