Question

Use the binomial series to expand the function as a power series. (1 − x)^1/5 state radius of convergence.

Answer 1

Valid only for |x|<1

Explanation:

Binomial expansion for rational powers is valid only if

|x|<1

If |x|<1 we have

`(1-x)^{(1)/(5) } =1+(1)/(5) (-x)+((1)/(5) (-4)/(5)x^2)/(2!) +((1)/(5) (-4)/(5) (-9)/(5)x^3 )/(3!) +...`

Same like integral powers except that instead of nCr we write here n(n-1)../r!

and there will be an infinite series

Thus we have

`(1-x)^{(1)/(5) } =1+(1)/(5) (-x)+((1)/(5) (-4)/(5)x^2)/(2!) +((1)/(5) (-4)/(5) (-9)/(5)x^3 )/(3!) +..\\= 1-(x)/(5) -(2x^2)/(25) +(12x^3)/(125) +...`