AP Precalc Unit 8 Notes

Sequences and series are fundamental concepts in mathematics that are related but distinct.

Sequence: A sequence is defined as an ordered list of numbers usually defined by some rule. Examples can include arithmetic sequences like (1, 2, 3, …) or geometric sequences like (2, 4, 8, …).
Series: A series is the sum of the terms of a sequence. Instead of listing terms with commas, series imply addition of terms, such as adding the first three terms of a sequence to yield a singular result.
- Formula representation: If a sequence has terms a1, a2, …, an, then the series can be written as: Sn = a1 + a2 + … + a_n

Difference Between Sequence and Series:
- A sequence provides individual terms, while a series aggregates these into a single value through addition.

When analyzing a sequence using a calculator, it is important to select the correct mode for calculations:
- Sequential mode allows users to define sequences more easily.
- Users may find direct entry of sequences more suitable rather than using procedural calculations.
- For example, entering a number sequence into the calculator results in output (e.g., 3, 5, 7, 9).

A recursive sequence is one where each term after the first is determined by applying a specific formula to the preceding term(s).
- Example: If the first term of the sequence is 4, the subsequent terms can be calculated as follows:
- Second term: 1st Term + 3 = 4 + 3 = 7
- Third term: 2nd Term + 3 = 7 + 3 = 10
- General representation: an = a{n-1} + 3
Fibonacci Sequence: This is another example of a recursive sequence where each term is the sum of the two preceding ones: Fn = F{n-1} + F_{n-2}

A partial sum refers to the summation of only certain terms within a series instead of all terms. For example, if we want to add just the first three terms of a series of 50:
- This technique is useful in mathematics to compute specific aggregates instead of entire series, helping simplify calculations.

Sigma (Σ) notation provides a concise way to represent a series of numbers to be summed and describes the range and specifics of the summation.
- Example of summation in sigma notation: ext{Sum} = ext{Σ}_{i=2}^{5} (2i^2)
Understanding sigma notation helps when calculating partial sums as it provides clarity on the number of terms involved and their corresponding indices.

Various mathematical properties help streamline work with series and sequences:
- Associative Property: This property allows you to regroup terms within the series without changing the outcome.
- Common Factor Property: C allows the factor to be pulled out and allows summation of the remaining terms separately, enhancing simplification.

The concept of infinite series can be challenging. For instance, dividing 7 by 9 yields $7/9$, which correlates with an infinite series but converges to a singular answer (e.g., repeating decimals).
Various techniques and properties can greatly assist in comprehending and calculating sequences and series effectively, especially in more complex problems involving recursion and summation.