How to prove the Generalised Binomial Theorem?

Question

At the end of the proof, the author says that the right hand side of the equation is $1$ when $x = 0$.

I think $0^k = 0$, then why does the sum equal to $1$?

@Rohan I read the post. I was, however, wondering the proof of binomial theorem when r is not positive integer, so the link did not answer fully. — qpzm, Dec 27 '17 at 07:38
To be frank the formula is $1+\sum_{k=1}^{\infty}\binom{r}{k}x^{k}$ and the first term $1$ is also included in the sigma notation for index $k=0$ based on a convention that $x^{0}=1$. — Paramanand Singh, Dec 27 '17 at 07:44

score 1 · Answer 1 · answered Dec 27 '17 at 08:02

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Definitions such as 0!=1, 0^0=1, and (r choose 0) =1 are conventionally applied to make sense of the first term in the summation.

answered Dec 27 '17 at 08:02

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And conventions such as 1!=1, x^1=x, (r choose 1)=r, etc. are conventionally applied to make sense of the subsequent terms. – bof Dec 27 '17 at 08:28

1 Answers1