The Alternating Series Test (AST) says than an alternating series converges if the [tex]u_n[\tex]'s are all positive, decreasing and its limit is 0. Does that exclude using it on terms that alternate every two or three terms? Even if it switches signs reliably?

Does this series converge or diverge?

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This series alternates sign every third term, but all the other conditions are met. If it doesn't satisfy the AST, then how would you determine it's convergence?