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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 10.4 : Convergence/Divergence of Series
For problems 1 – 4 compute the first 3 terms in the sequence of partial sums for the given series.
- \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{1}{{1 + {3^n}}}} \)
- \( \displaystyle \sum\limits_{n = 1}^\infty {\left( {{2^n} - {3^n}} \right)} \)
- \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{1 + n}}{{2n}}} \)
- \( \displaystyle \sum\limits_{n = 0}^\infty {10} \)
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.
- \({s_n} = \left( {{n^2} + 4n} \right){{\bf{e}}^{ - 2n}}\)
- \(\displaystyle {s_n} = \frac{{1 + 2n + 3{n^2}}}{{4{n^2} + 5n + 6}}\)
- \(\displaystyle {s_n} = \frac{n}{{\ln \left( {n + 2} \right)}}\)
- Let \(\displaystyle {d_n} = \frac{{7 - 8n}}{{4 + 3n}}\)
- Does the sequence \(\left\{ {{d_n}} \right\}_{n = 0}^\infty \) converge or diverge?
- Does the series \( \displaystyle \sum\limits_{n = 0}^\infty {{d_n}} \) converge or diverge?
- Let \({d_n} = 1 + n{{\bf{e}}^{ - n}}\)
- Does the sequence \(\left\{ {{d_n}} \right\}_{n = 0}^\infty \) converge or diverge?
- Does the series \( \displaystyle \sum\limits_{n = 0}^\infty {{d_n}} \) converge or diverge?
For problems 10 – 12 show that the series is divergent.
- \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{9 - 2{n^2}}}{{1 + 4n + {n^2}}}} \)
- \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{{{5^n} + 1}}{{{3^n}}}} \)
- \( \displaystyle \sum\limits_{n = 1}^\infty {\cos \left( n \right)} \)