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### Section 10.4 : Convergence/Divergence of Series

For problems 1 & 2 compute the first 3 terms in the sequence of partial sums for the given series.

1. $$\displaystyle \sum\limits_{n = 1}^\infty {n\,{2^n}}$$ Solution
2. $$\displaystyle \sum\limits_{n = 3}^\infty {\frac{{2n}}{{n + 2}}}$$ Solution

For problems 3 & 4 assume that the $$n$$th term in the sequence of partial sums for the series $$\displaystyle \sum\limits_{n = 0}^\infty {{a_n}}$$ is given below. Determine if the series $$\displaystyle \sum\limits_{n = 0}^\infty {{a_n}}$$ is convergent or divergent. If the series is convergent determine the value of the series.

1. $$\displaystyle {s_n} = \frac{{5 + 8{n^2}}}{{2 - 7{n^2}}}$$ Solution
2. $$\displaystyle {s_n} = \frac{{{n^2}}}{{5 + 2n}}$$ Solution

For problems 5 & 6 show that the series is divergent.

1. $$\displaystyle \sum\limits_{n = 0}^\infty {\frac{{3n\,{{\bf{e}}^n}}}{{{n^2} + 1}}}$$ Solution
2. $$\displaystyle \sum\limits_{n = 5}^\infty {\frac{{6 + 8n + 9{n^2}}}{{3 + 2n + {n^2}}}}$$ Solution