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Section 10.11 : Root Test

For each of the following series determine if the series converges or diverges.

1. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{{{2^{1 - 3n}}}}{{{{\left( { - 3} \right)}^{1 - 2n}}}}} \)
2. \( \displaystyle \sum\limits_{n = 2}^\infty {{{\left( {\frac{{5{n^2} - 2n + 1}}{{3{n^2} + n - 3}}} \right)}^{ - n}}} \)
3. \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{{{10}^{3 + 5n}}}}{{{n^{1 + {n^2}}}}}} \)
4. \( \displaystyle \sum\limits_{n = 1}^\infty {{{\left( {\frac{{6 - 9n}}{{2 + 4n}}} \right)}^{\frac{1}{3}n}}} \)