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Find a power series representation for the function and determine the radius of convergence.

$ f(x) = \ln (5 - x) $

$\ln 5-\frac{1}{5^{n+1}} \sum_{n=0}^{\infty} \frac{x^{n+1}}{n+1}$

$R=5$

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Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

fun power, serious reputation for function F ax and determine the readers of convergence. Okay, so I'm gonna transport axe into lawn with this pool of five, and it's because against one ministers or five And what this This is long fine, plus lawn one minus X or if I and we're going to spend this part, actually, what is that? So, uh, this is just the integral of when there were one minute six or five the X. So let's have not. Maybe we should modify this by some constant K here, so let's try this. So, what is the Negro Levitt? Is this on long one minus plus to power? That's over five. And I think there is some constant here, so Okay, let's do this. So, what is the relative of long one minus? That's or fight. So this is one hour four, five times, not people one thing. Okay, so we confirmed that the way it goes, Nothing. One fifth. So, uh, yeah, okay. And actually, we could spend this part. So one of our women, this X for fighting is very easy to be expanded into hair Siri's. So this particle becomes to end from zero could be and, uh, extra power in your health. And here I have the differential of reader, the X. So we're gonna integrate this one through the term and actually were able to change the other integral. And the summation in there on his son. So on with the first girl extra card and ORF of love in the X and this summit. And this is just equal to our own life. By my eyes, when you were to the power from past one and some notation Inigo is explored in the eggs. It's rather familiar to us. So on. Okay, let's first you write this part and this is just lawn five minus when there are servants, US one and the integral is gonna be extra power and plus one over and us one. So that's the girl. Yes. Okay, so now we have our results. That's the power of serious reputation for the function for FX on determine the readers of Convergence. So how to find the richest convergence? We gotto push up on it from the step where we just expand dysfunction. So it's this equation we require that that's or fine is from zero ice from the ones who want question, place, axis from knocking fight to five. So that's the rays of convergence, okay?

University of Illinois at Urbana-Champaign