Question: There are a variety of tests used to determine the convergence of infinite series.

Task:

1. Use the comparison test to determine the convergence or divergence of the series
   \[4+\frac{1}{5}+0.3+\frac{1}{3+\sqrt{2}}+\frac{1}{9+\sqrt{3}}+\frac{1}{27+\sqrt{4}}+\frac{1}{81+\sqrt{5}}+...\] , showing all work.
2. Use the alternating series test to show the convergence or divergence of the series
   \[{{\sum\limits_{i=1}^{\infty }{(-1)}}^{(i+1)}}\frac{i+3}{{{i}^{2}}+10}\] , showing all work.
3. Determine whether the \[{{\sum\limits_{j=1}^{\infty }{(-1)}}^{(j+1)}}\frac{j+3}{j+10}\] converges absolutely, converges conditionally, or diverges, showing all work.
4. Use the ratio test to determine the convergence or divergence of the series defined by
   \[{{a}_{1}}=10,{{a}_{n+1}}=\frac{5}{n}{{a}_{n}}\] , showing all work.
5. Use the root test to determine the convergence or divergence of the series \[\sum\limits_{n=1}^{\infty }{\frac{100{{n}^{2}}}{{{e}^{n}}}}\] showing all work.

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Deliverable: Word Document