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### Section 4-10 : Ratio Test

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

1. $$\displaystyle \sum\limits_{n = 1}^\infty {\frac{{{3^{1 - 2n}}}}{{{n^2} + 1}}}$$ Solution
2. $$\displaystyle \sum\limits_{n = 0}^\infty {\frac{{\left( {2n} \right)!}}{{5n + 1}}}$$ Solution
3. $$\displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 2} \right)}^{1 + 3n}}\left( {n + 1} \right)}}{{{n^2}{5^{1 + n}}}}}$$ Solution
4. $$\displaystyle \sum\limits_{n = 3}^\infty {\frac{{{{\bf{e}}^{4n}}}}{{\left( {n - 2} \right)!}}}$$ Solution
5. $$\displaystyle \sum\limits_{n = 1}^\infty {\frac{{{{\left( { - 1} \right)}^{n + 1}}}}{{6n + 7}}}$$ Solution