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Section 10.7 : Comparison Test/Limit Comparison Test

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

1. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{{{3^n} + n}}{{{2^{n + 1}}}}} \)
2. \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{4n - 3}}{{2{n^5}}}} \)
3. \( \displaystyle \sum\limits_{n = 4}^\infty {\frac{1}{{\left( {2n - 1} \right)\left( {n - 3} \right)}}} \)
4. \( \displaystyle \sum\limits_{n = 8}^\infty {\frac{{\ln \left( {{n^2}} \right)}}{n}} \)
5. \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{4n}}{{{{\left( {n + 1} \right)}^3}}}} \)
6. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{{n - 4}}{{\left( {{n^2} + 1} \right){{\bf{e}}^n}}}} \)
7. \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{\sqrt {2 + \cos^2\left( {5n} \right)} }}{{\sqrt {{n^2} - n - 1} }}} \)
8. \( \displaystyle \sum\limits_{n = 3}^\infty {\frac{{n - 1}}{{\sqrt {{n^3} + n + 3} }}} \)
9. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{{3{n^2} + 7n - 1}}{{{n^4} - n + 3}}} \)
10. \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{\left( {1 - \sin \left( n \right)} \right)\left( {1 + \sin \left( n \right)} \right)}}{{{n^2} + 8n + 1}}} \)