NJC Sequence and Series Tutorial Q1

For each of the following, describe the behaviour of the sequence and determine whether the sequence is convergent or divergent. Find the limit of the sequence if it converges.

1. \(\left\{ \frac{1}{4}-\frac{2}{{{\left( 3n+1 \right)}^{2}}} \right\},n\in {{\mathbb{Z}}^{+}}\)
2. \(\left\{ \frac{5{{n}^{2}}+4n+1}{\left( 2n+1 \right)\left( 3n-2 \right)} \right\},n\in {{\mathbb{Z}}^{+}}\)
3. \(\left\{ {{\left( -2 \right)}^{n}} \right\},n\in {{\mathbb{Z}}^{+}}\)
4. \(\left\{ {{\left( \frac{1}{2} \right)}^{n}} \right\},n\in {{\mathbb{Z}}^{+}}\)