The given series is \( \sum_{n=1}^{\infty}\left(\frac{-2 n}{n+1}\right)^{3 n} \).
The root test states that for a series \( \sum_{n=1}^{\infty} a_n \), if \( \lim_{n \to \infty} \sqrt[n]{|a_n|} = L \), then:
- If \( L < 1 \), the series converges.
- If \( L > 1
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