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Section 10.8 : Alternating Series Test

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

1. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{{{{\left( { - 1} \right)}^{n + 7}}}}{{{n^2} + 3}}} \)
2. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 2}}}}{{{3^n} + 3n}}} \)
3. \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 1}}}}{{{n^3} + 4{n^2} + 8}}} \)
4. \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{1}{{{{\left( { - 2} \right)}^n}\left( {6n + 1} \right)}}} \)
5. \( \displaystyle \sum\limits_{n = 3}^\infty {\frac{{4n\cos \left( {n\pi } \right)}}{{2{n^2} + 1}}} \)
6. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 10}}\,\,{n^2}}}{{{n^3} + {n^2} + 4}}} \)
7. \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{{{\left( { - 1} \right)}^{n + 5}}\left( {2n + 1} \right)}}{{{n^2} + 8}}} \)