Question

Find the first four non-zero terms in the binomial expansion of


`(1 + (x)/(125) )^{ (1)/(3) }`
in ascending powers of

`x`

Answer 1

`\displaystyle \left(1+(x)/(125)\right)^{(1)/(3)}=\sum\limits_(n=0)^(\infty)\binom{(1)/(3)}{n}\left((x)/(125)\right)^(n)\\\\ \approx 1+(\left((1)/(3)\right))/(1!)\left((x)/(125)\right)+(\left((1)/(3)\right)\left(-(2)/(3)\right))/(2!)\left((x)/(125)\right)^(2)+(\left((1)/(3)\right)\left(-(2)/(3)\right)\left(-(5)/(3)\right))/(3!)\left((x)/(125)\right)^(3)\\\\ \approx 1+(1)/(375)x-(1)/(140625)x^(2)+(1)/(31640625)x^(3)`