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Suppose that $ \sum_{n = 0}^{\infty} c_nx^n $ converges when $ x = - 4 $ and diverges when $ x = 6. $ What can be said about the convergence or divergence of the following series?

(a) $ \sum_{n = 0}^{\infty} c_n $

(b) $ \sum_{n = 0}^{\infty} c_n8^n $

(c) $ \sum_{n = 0}^{\infty} c_n( - 3)^n $

(d) $ \sum_{n = 0}^{\infty} ( - 1)^n c_n 9^n $

a) Converges

b) divergent

c) Converges

d) Diverges

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Missouri State University

Campbell University

University of Michigan - Ann Arbor

okay, If we converge when X is equal to minus four and we diverge when X is equal to six, then we know that we for sure going to converge in this interval, right? We could converge in more places than that, but we're at least gonna converge here. Our radius of convergence is less than or equal to six. So best case scenario, our radios have convergence would be six. But it we can't possibly have a radius of convergence bigger than six because we get divergence when X is equal to six. So hear this corresponds tow this sum when X is equal to one. So one is inside of this interval of convergence. So we're going to get convergence for a for be This corresponds to ax equals eight so that outside of where we would be converging here eight is bigger than six. So we get divergence for sea. See Corresponds to this With X being minus three minus three is in here. So we get convergence and for D. This corresponds to this sum when x is minus nine. Minus nine is nine and absolute value, which is bigger than our radius of convergence is so we can't possibly get convergence for Dean