# 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

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?

• converges to $$\dfrac{3}{2}$$
• converges to $$\dfrac{18}{509}$$