Section 12.1: Sequences and Summation Notation

1. Recognize the notation \(a_{n} := a(n)\) for a sequence as a function whose domain is the natural numbers \(n \in \mathbb{N}\), i.e. \(n\) is one of the ‘counting numbers’ \(1, 2, 3, \ldots\).
2. Given a closed-form expression for a sequence \(a_{1}, a_{2}, \ldots, a_{n}, \ldots\), find the first \(N\) terms of the sequence.
3. Given several terms of a sequence that can be expressed in a closed-form, determine the closed-form expression for the sequence.
4. Explain what it means for a sequence to be recursively defined.
5. Given a recursively defined sequence, determine the first \(N\) terms of the sequence.
6. Find the \(n\text{th}\) partial sum \(S_{n}\) of a sequence.
7. Recognize the \(n\text{th}\) partial sum \(S_{1}, S_{2},\ldots, S_{n}, \ldots\) of a sequence as itself a sequence.
8. Recognize and use summation notation, aka Sigma notation, aka \(S_{n} = \displaystyle\sum_{k = 1}^{n} a_{k}\), for the \(n\text{th}\) partial sum \(S_{n}\).
9. Use sequences and partial sums to model situations that require discrete changes to some quantity.