Issue with Binomial Expansion Formula

[Reingley]

Working through Leonard Susskind's book The Theoretical Minimum, I noticed an issue with his expansion for the Binomial Expansion (he was missing factorials in the denominators). This led me to some confusion about the final term that is generally written (bⁿ).

(a+b)ⁿ = aⁿ + na^n-1b + n(n-1)/2! a^n-2b² + ... + bⁿ

My issue with this is that if one were to solve for n = 2, the b² term comes out from the 3rd term (2! term) and there is no need to add the bⁿ⁼² term at the end.

Is my problem that if I am using n = 2, I shouldn't even bother to include the 3rd term (and all higher order terms that incidentally go to zero anyways)? It seems to me that the bⁿ term is simply unnecessary at the end.

Thanks for any input!

 

[fresh_42]

For $n=2$ the third term and $b^2$ are identical. So the notion of $b^n$ is there to show where the expansion ends.
Of course you are free to write it as $\binom{n}{n}a^{n-n}b^n$ instead, but $b^n$ is more convenient.

In its closed version $(a+b)^n = \sum_{k=0}^n \binom{n}{k}a^{n-k}b^k$ the last term is written the way you want it to be.

 

[Reingley]

Thanks! That makes sense. I figured it was confusion on my part with the notation, but I wanted to check that I hadn't overlooked something obvious.