[Math] Why is the binomial expansion not valid for an irrational index

Question

I recently learned about the binomial theorem for any index at my school. The index was explicitly mentioned to belong to the set of rational numbers. My instructor didn't give us a proof to back this statement, but rather just said that the actual proof of the expansion requires the index to be rational. This wasn't a problem, but it soon became one. I was thinking about what would happen if I used the expansion for a binomial having a rational index on a binomial having an irrational index. Surprisingly, the directly calculated value was extremely close to the value which was calculated using the expansion.

As can be seen, the result is correct up to 5 decimal places despite only using the first four terms out of the infinite terms. I am confused now, as to why did this happen. Any explanation would be highly appreciated.

Thanks

Answer 1

The binomial theorem is valid for real indices in general, even irrational ones. Possibly what your instructor meant to say is that there is a nice elementary proof that only holds for rational indices.