Question

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

Answer 1

Answer:

Valid only for |x|<1

Step-by-step explanation:

Binomial expansion for rational powers is valid only if

|x|<1

If |x|<1 we have

$(1-x)^{\frac{1}{5} } =1+\frac{1}{5} (-x)+\frac{\frac{1}{5} \frac{-4}{5}x^2}{2!} +\frac{\frac{1}{5} \frac{-4}{5} \frac{-9}{5}x^3 }{3!} +...$

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

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