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Section 10.5 : Special Series

For each of the following series determine if the series converges or diverges. If the series converges give its value.

1. \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{ - 2}}{{{n^2} + n}}} \)
2. \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{12}}{n}} \)
3. \( \displaystyle \sum\limits_{n = 1}^\infty {{5^{n + 3}}\,{4^n}} \)
4. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{3}{{{4^{n + 1}}\,{5^{1 - n}}}}} \)
5. \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{1}{{14\,n}}} \)
6. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{7}{{{n^2} + 5n + 6}}} \)
7. \( \displaystyle \sum\limits_{n = 1}^\infty {{4^{1 + 2n}}\,{3^{2 - 3n}}} \)
8. \( \displaystyle \sum\limits_{n = 4}^\infty {{4^{1 + 2n}}\,{3^{2 - 3n}}} \)
9. \( \displaystyle \sum\limits_{n = 3}^\infty {\frac{5}{{{n^2} - 1}}} \)
10. \( \displaystyle \sum\limits_{n = 4}^\infty {\frac{1}{{{n^2} - 4n + 3}}} \)
11. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{{{5^{3 + n}}}}{{{2^{2 + 3n}}}}} \)