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Section 4-4 : 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