power serie

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power serie

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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\[\sum_{n=1}^{\infty} (-1)^{n+1}*\frac{ (x-1)^n }{ n }\] convergence interval a. (-1,1) b.[0,2) c. (0,2] d.[0,2] e.[-1,1]
hmm you are expanding around 1, so disregard answer a and e
radius of convergence is 1, so it is one of the middle three your job is to check at the endpoints, and see if it converges at \(x=0\) and if it converges at \(x=2\)

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do you know how to do that?
no, i don't
lets replace \(x\) by 2
the terms will look like \[(-1)^{n+1}\frac{(2-1)^n}{n}=(-1)^{n+1}\frac{1}{n}\]
this is an alternating series, whose terms go to zero, and so when you sum it, it will converge
now we repeat the process with \(x=0\) but before we do, is what i wrote above clear?

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