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### Section 4-13 : Estimating the Value of a Series

1. Use the Integral Test and $$n = 10$$ to estimate the value of $$\displaystyle \sum\limits_{n = 1}^\infty {\frac{n}{{{{\left( {{n^2} + 1} \right)}^2}}}}$$. Solution
2. Use the Comparison Test and $$n = 20$$ to estimate the value of $$\displaystyle \sum\limits_{n = 3}^\infty {\frac{1}{{{n^3}\ln \left( n \right)}}}$$. Solution
3. Use the Alternating Series Test and $$n = 16$$ to estimate the value of $$\displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^n}n}}{{{n^2} + 1}}}$$. Solution
4. Use the Ratio Test and $$n = 8$$ to estimate the value of $$\displaystyle \sum\limits_{n = 1}^\infty {\frac{{{3^{1 + n}}}}{{n\,{2^{3 + 2n}}}}}$$. Solution