Review of Sequences and Series

Review of Sequences and Series

Definitions and Notations

  • Sequence: A function that has its domain as either positive or non-negative integers.
  • Example of a sequence:
    • a_n = {1, 2, 3, 4, 5, 6, 7}
    • b_n = {3, 8, -2, 8, 73}
  • Note: A sequence has no zero value in positive sequences.

Types of Sequences

  • Arithmetic Sequence: A sequence where the difference between consecutive terms is constant.
    • Example: 1, 2, 3, \ldots
  • Geometric Sequence: A sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Alternating Sequences

  • Example: (-1)^n
    • which denotes alternating signs.
    • Pattern: 1, -1, 1, -1, \ldots

Factorials and Their Applications

  • Definition: The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n.
  • Formula: n! = n \times (n-1) \times (n-2) \times \cdots \times 3 \times 2 \times 1
  • Special case: 0! = 1
  • Example Calculation for Factorials:
    • 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
    • 4! = 4 \times 3 \times 2 \times 1 = 24
    • For n=7: 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040

Recursive Sequences

  • Definition: A sequence defined by a recurrence relation, where each term is formulated based on previous terms.
  • Example: Fibonacci sequence defined as:

    • Fn = \begin{cases} 1 & \text{if } n=1 \ 1 & \text{if } n=2 \ F{n-1} + F_{n-2} & \text{if } n > 2
      \end{cases}
  • Produces series: 1, 1, 2, 3, 5, 8, 13, 21, \ldots

Sigma Notation for Series

  • Definition: A compact way to represent a sum of a sequence.
  • Example: \Sigma{i=1}^{n} ai represents the sum of the sequence where a_i are terms of the sequence.

Example Calculations

  • Calculation of Series:
    • Given 2 + 5 + 7 + \ldots up to the 10th term can be expressed in summation.
  • Another example: Calculate values for different function applications like:
    • If h = 4, then evaluate expressions involving factorials and simple arithmetic operations, such as:
    • \frac{4! \cdot 3}{2^5}

Combinatorial Sequences

  • Combinatorial Factors: Mean interactions and arrangements in a sequence where combinations are needed.
  • Formula for combinations (n choose k): C(n, k) = \frac{n!}{k!(n - k)!}.

Numerical Data and Statistics

  • Examples of Series Summation:
    • Example provided: Calculate values recursively or through direct application of functions.
    • Evaluate sequences using specific number patterns like:
    • 3h + 1
    • h = { \text{specific values ranging up to 10} }
    • Apply these in geometric and arithmetic contexts.

Conclusion

  • Understanding sequences and series is crucial in mathematical studies, especially in calculus and statistics.
  • The applications of sequences extend to various fields including computer science (algorithm analysis), finance (interest computations), and combinatorics (counting problems).