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Numerade Educator

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Problem 31 Hard Difficulty

Use the binomial series to expand the function as a power series. State the radius of convergence.

$ \sqrt [4]{1 - x} $

Answer

$1-\frac{1}{4} x-\sum_{n=2}^{\infty} \frac{3 \cdot 7 \cdots \cdots(4 n-5)}{4^{n} \cdot n !} x^{n}, \quad R=1$

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Video Transcript

So we're gonna want Thio. Use the binomial Siri's to expand the function as a power. Siri's eso To do that, we're going to restate this right here. One minus X is one minus X to the 1/4 um and then an even better way for us to write. That would be one plus x r one plus a negative X to the 1/4. So with that, we see that the binomial Siri's we know is one plus X to the K is equal to the summation of an equal zero to infinity. Then we have the K over end. X to the end is equal to one plus k x plus Hey, times K minus one over two factorial x squared plus que times k minus one times K minus two over three factorial. And then that's gonna be X cubed and that's going to keep going. So when we plug in a negative acts instead of just a positive X on DWI, let K equal 1/4 where we're gonna end up getting. Is this right here 1/4 and negative X to the end. That's gonna equal one minus 1/4 X minus three over four times four times two factorial X squared minus three times seven over. Four times four times four times three factorial minus three times seven times 11 over four to the fourth times four factorial x to the fourth on the Sumi X cubed. And that's gonna keep going on with subtraction. So another way that we could rewrite this is one minus one over for X minus the summation from an equals two to infinity of three times seven times four and minus five. That's all over four to the end. My times and factorial times x two, then that is our general way of writing it. Um, this part right here is saying that there are numbers between three and four and minus five. Um, so this is easier to use in the ratio test. So to find the ratio of convergence, um, we'll do the ratio test, which is absolute value of a and plus one over a N. So when we do that, we end up getting four and minus one over four times n plus one x, and we take the limit of that and what the limit ends up being If we take the limit as n goes to infinity and we do the same thing here, what will end up getting is just the absolute value of X. So the result is the same. Um, what we end up getting is that it converges if and only if X is less than the absolute value of X is less than one. So we know that our radius of convergence is equal toe one.