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Graph the first several partial sums $ s_n (x) $ of the series $ \sum_{n = 0}^{\infty} x^n, $ together with the sum function $ f(x) = 1/(1 - x), $ on a common screen. On what interval do these partial sums appear to be converging to $ f(x)? $

(-1,1)

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Campbell University

Harvey Mudd College

University of Nottingham

Boston College

All right, So we'LL just start with s one of X, which is x zero plus X to the one. So one plus acts, which is something as x plus one. So this is just a line shifted up One unit. Okay. And the graph of one over one minus X. It's flat over here. And then it blows up when x is equal to one, and then it's negative over here. Okay. And over here, this is this is that X equals minus one. Okay, so it looks like we're already running into bad news once we get past X equals one, because F of X is going to be negative, whereas our partial some is going to be positive. Okay, so that that could change. We've got to keep looking. So ask two of X. This is X squared plus X plus. One case is going to look and like a quadratic here, and then graphing affects again. We get you know this thing. Good. Okay. And again, we see. Looks like bad news is happening at one at minus one. As two of minus one is going to be positive one. So here it was. Zero. Here It's positive One. Okay. And then we can look at as three of axe. So that's X cubed plus X squared, plus tax plus one. So this is going to look something like this. And then the graph of f of axe again is, you know, well, this should just be flattening here. Graph of death of ex blows up. Want to get to one over at X equals minus one. If we plug minus one into here we get Ah, minus one plus one minus one plus one will get zero. So notice that here. Zero. Here we get minus one, and then we're backto one. When we plug in, X equals minus one. So looks like there's some bad news happening. X equals minus one as well. Since we appear to be oscillating well, when we should include this other part of f of X. What vexes past one. Then we get negative stuff here. Whereas our partial some here is just going to be positive when we're past one. Okay, so we ran into the same problem with these partial sums as well here. This partial sun was positive when exit past one this partial. Some would be positive when exit past one. But again, after vex is going to be negative for those. So at X equals one run into some bad news at X equals minus one. We run into some bad news, but between minus one and one, these guys look pretty similar to each other, so it looks like we get convergence between minus one and one.