Binomial Series — Question 8

*Question 8 Let $m$ be a nonnegative integer. Explain algebraically why the generalized binomial series for $(1+x)^m$ terminates, and recover the ordinary finite Binomial Theorem. Illustrate with $m=4$.

*Question 8 – Solution

$$\binom{m}{n}=\frac{m(m-1)\cdots(m-n+1)}{n!}.$$ If $n=m+1$, the numerator includes the factor $m-m=0$. For every $n>m$, it still includes that zero factor. Hence $$\binom{m}{n}=0\qquad(n>m).$$

The formally infinite series therefore reduces to $$\boxed{(1+x)^m=\sum_{n=0}^{m}\binom{m}{n}x^n}.$$ Because this is a polynomial identity, it is valid for every real (and complex) $x$; there is no finite radius restriction.

$$(1+x)^4=1+4x+6x^2+4x^3+x^4.$$ The next coefficient is $\binom45=0$, so no higher powers occur.