Find radius of convergence for <munderover> &#x2211;<!-- ∑ --> <mrow class="MJX-TeXAtom-

studovnaem4z6 2022-05-16 Answered
Find radius of convergence for
n = 1 ( x 1 ) n 3 + ( x 1 ) n 1 4 n + 2 2 n 1
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Answers (1)

Calvin Oneill
Answered 2022-05-17 Author has 20 answers
The way it's written, it's a weird series to ask for radius of convergence, as it has a singularity at x = 1
Forgetting about that, Alex Becker's answer is spot on: to find the radius of convergence you take the limit over n, not x, and only of the coefficients: here it would be
R = lim n a n a n + 1 = lim n ( 1 4 ) n ( 1 4 ) n + 1 = 4.
Or, if you want to go for understanding instead of just methods, again as Alex said this is a geometric series so it will converge when | x 1 | / 4 < 1, i.e.
| x 1 | < 4.
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