Series practice

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


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.

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.

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?

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

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

Answers (mixed up)