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Section 4-4 : Convergence/Divergence of Series

For problems 1 – 4 compute the first 3 terms in the sequence of partial sums for the given series.

  1. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{1}{{1 + {3^n}}}} \)
  2. \( \displaystyle \sum\limits_{n = 1}^\infty {\left( {{2^n} - {3^n}} \right)} \)
  3. \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{1 + n}}{{2n}}} \)
  4. \( \displaystyle \sum\limits_{n = 0}^\infty {10} \)

For problems 5 – 7 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. \({s_n} = \left( {{n^2} + 4n} \right){{\bf{e}}^{ - 2n}}\)
  2. \(\displaystyle {s_n} = \frac{{1 + 2n + 3{n^2}}}{{4{n^2} + 5n + 6}}\)
  3. \(\displaystyle {s_n} = \frac{n}{{\ln \left( {n + 2} \right)}}\)
  4. Let \(\displaystyle {d_n} = \frac{{7 - 8n}}{{4 + 3n}}\)
    1. Does the sequence \(\left\{ {{d_n}} \right\}_{n = 0}^\infty \) converge or diverge?
    2. Does the series \( \displaystyle \sum\limits_{n = 0}^\infty {{d_n}} \) converge or diverge?
  5. Let \({d_n} = 1 + n{{\bf{e}}^{ - n}}\)
    1. Does the sequence \(\left\{ {{d_n}} \right\}_{n = 0}^\infty \) converge or diverge?
    2. Does the series \( \displaystyle \sum\limits_{n = 0}^\infty {{d_n}} \) converge or diverge?

For problems 10 – 12 show that the series is divergent.

  1. \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{9 - 2{n^2}}}{{1 + 4n + {n^2}}}} \)
  2. \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{{{5^n} + 1}}{{{3^n}}}} \)
  3. \( \displaystyle \sum\limits_{n = 1}^\infty {\cos \left( n \right)} \)