Binomial Approximation

$(a+b{)}^n\approx a^n\left(1+\frac{bn}{a}\right)\text{ if }\left|\frac{bn}{a}\right|\ll 1$

$(1+b{)}^n\approx 1+bn$

• Setting previous equation’s $a=1$ gives this one which is just the first two terms of the Taylor expansion around $b=0$
  • The next few terms would be $\frac{1}{2}n(n-1)b^2+\frac{1}{6}n(n-1)(n-2)b^3+\frac{1}{24}n(n-1)(n-2)(n-3)b^4+O(b{)}^5$, where $O(b{)}^5$ is a polynomial of order $5$ or greater
  • Shown by Mathematica code:

Series[(1+b)^n,{b,0,4}]