Series practice

When answering the following, keep in mind series and series tests that we have discussed …

• "nth term" divergence test
• telescoping
• geometric
• integral test
• p-series
• direct comparison
• limit comparison

Tasks

1. Determine whether the series \( \displaystyle\sum_{n=1}^{\infty} \dfrac{2}{n^2 + 2n}\) is convergent or divergent by expressing the partial sums, \(S_N\), as a telescoping sum. If it is convergent, find its sum.

hint 1

hint 2

2. Test the series \( \displaystyle\sum_{m=1}^{\infty} \dfrac{6^{m+1}}{4^{5m}} \) for convergence or divergence. If the series is geometric and convergent, find its sum.

hint 1

hint 2

3. Is the series \( \displaystyle\sum_{n=1}^{\infty} \dfrac{n^3 + 2n^2 + 3n + 4}{n^4 + 3n^3 + 2n^2 + n} \) convergent or divergent?

hint 1

hint 2

4. Determine convergence or divergence of \( \displaystyle\sum_{n=3}^{\infty} \dfrac{1}{e^{0.5 n}} \)

hint

alt

5. Does \( \displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^n} \) converge or diverge?

hint 1

hint 2

alt

Answers (mixed up)

• converges to \( \dfrac{3}{2} \)
• diverges by limit comparison
• converges by direct comparison
• converges to \( \dfrac{18}{509} \)
• converges by integral test