This is for my qualifying exam prep. It looks like both ratio and root tests failed because their limits do not exist. Any help will be appreciated.
2026-04-13 12:06:29.1776081989
Find the interval of convergence of the series $\sum_{n=0}^\infty ((-1)^n+3)^n(x-1)^n$.
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Using the root test with $a_n=((-1)^n+3)^n$ we have $\sqrt[n]{|a_n|}=(-1)^n+3$. $\limsup_n\sqrt[n]{|a_n|}=4$. Thus the series converges for $|x-1|<\frac{1}{4}$