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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.

Here are the calculations that I did

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

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    $\begingroup$ There is, in fact, a generalized binomial theorem allowing for real, and even complex exponents, using an infinite series. You can even allow $x$ and $y$ to be complex. The proof is more complicated, though, which is likely why your instructor didn't touch on it. en.wikipedia.org/wiki/Binomial_theorem#Generalizations $\endgroup$
    – Kevin Long
    Commented Dec 20, 2016 at 19:28

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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.

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