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

Question

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

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Adalyn 2 weeks 2021-10-12T08:02:28+00:00 1 Answer 0

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    2021-10-12T08:04:02+00:00

    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} +...

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