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Problem 14 Medium Difficulty

(a) Use Equation 1 to find a power series representation for $ f(x) = \ln (1 - x). $ What is the radius of convergence?
(b) Use part (a) to find a power series of $ f(x) = x \ln(1 - x). $
(c) By putting $ x = \frac {1}{2} $ in your result from part (a), express $ \ln 2 $ as the sum of an infinite series.


A. $\sum_{n=1}^{\infty}-\frac{x^{n}}{n}, \quad R=1$
B. $$x \ln (1-x)=-\sum_{n=0}^{\infty} \frac{x^{n+2}}{n+1}$$
C. $\sum_{n=1}^{\infty} \frac{1}{n 2^{n}}$


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Video Transcript

Okay, let's get started on first. Find the power. Serious reputation for ethics was long of one minus X. Okay, we're gonna start with very familiar terrace, parent of these function. So this is equals to Seema from a and zero two, if any extra help in. And ah, that's fuzzy relation between a state desist re x. So either any relation between African Jax. It is not hard for fun that the derivative of X equals two. One more witness X Times ninety one. So this equals to Ministry of X. Okay, so Well, this the pain expanding for Ministry of X, that is just ninety one times and from zero to infinity extra power in, and we're going in to grow it. This I don't think we're going anywhere right by from both sides and that this is going to be so here is in the grow, and we can change the other Symon and he grew. So there's going to be not one times signal from zero to infinity and except in class one or And this one Yes, on DH. This is just the so here. Well, there's the anywhere of minus jaxx. This isjust fx So we conclude that the tailor experience FX equals to ninety one times and from zeros for infinitely extra towers in past one and plus one. This is just no, you won hundreds except off in the war in and from one two even. Okay and next. So use party to fund the power Siri's of wrath of X equals X times. So Okay, so this part, let's say this is the two of us and this is one of X. So after two of its just equals two x times of one backs. So it's going to be ninety one homes and from one to infinity x two power and plus one or in and see, but putting at least one half your result from part, eh? Expressing onto as some of you in Siri's. Okay, so all this lawn, too, so long to ecos, too long of one minus minus one. Okay, actually wait needs food. Defined the readers of convergence for one of ax so worthy way Expanded Serious is from the bear familiar. Siri's one of the woman's acts and the rudest converters for this service is just the absolute value is less than one. So Yeah, we're just talking one minus ninety one this long to into every one of X. So this is just I've won the minus one, and they're looking to get for me. We can get so ah, we're plugging one here. So it's going to be because not few times. So if this is one and this minus one three powers in tell her one. So Okay, so basically, it's just a that you won class half of not You went to the park to its one and the miners when? Third class, one fourth. And so on. Um, minus plus, not you wanted power. And over in. So yet this is our result. And we can simply fight as one minus one half plus one third months. One fourth and plus, not you want to earth and plus one in minus one equals to invest one street in the sense of the the value of minus one to the power off That's over, over in. So yeah, this is our results. Okay?

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