I am finding if the sequence $$(-1)^n\frac{(n+1)!}{3n!}$$ is convergent or divergent. I want to clarify that this sequence is divergent since the Squeeze theorem cannot be applied here (because there is (n+1)! in the numerator) and when simplified, the sequence yields to {$(-1)^n\frac{(n+1)}{3}$}; thus, the sequence does not approach to a single value. Am I right?

  • 3
    $\begingroup$ Are you sure you have $(3)(n!)$ and not $(3n)!$ in the denominator? $\endgroup$ Jun 3 '21 at 11:50
  • 1
    $\begingroup$ Yes. The term in the denominator is not enclosed in a parenthesis. $\endgroup$
    – Fubuki
    Jun 3 '21 at 12:00
  • 2
    $\begingroup$ Correct. The sequence is unbounded, so it cannot have a limit. The series emerging by summing up the entries does not converge either. $\endgroup$
    – Peter
    Jun 3 '21 at 12:13

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