Key Concepts of Sequences to Know for Calculus II

Sequences are ordered lists of numbers that play a key role in calculus. They can converge to a limit or diverge, impacting how we analyze functions and series. Understanding sequences helps us grasp the foundations of limits and infinite series.

1. Definition of a sequence

   • A sequence is an ordered list of numbers, typically defined by a function or a rule.
   • Each number in the sequence is called a term, and is usually indexed by natural numbers.
   • Sequences can be finite (having a limited number of terms) or infinite (continuing indefinitely).

2. Convergence and divergence of sequences

   • A sequence converges if its terms approach a specific value (the limit) as the index goes to infinity.
   • A sequence diverges if it does not approach any finite limit; it may go to infinity or oscillate.
   • Understanding convergence is crucial for determining the behavior of sequences in calculus.

3. Limit of a sequence

   • The limit of a sequence is the value that the terms of the sequence get closer to as the index increases.
   • If a sequence converges, the limit can be found using various techniques, such as algebraic manipulation or the squeeze theorem.
   • The notation for the limit of a sequence is often expressed as ( \lim_{n \to \infty} a_n = L ).

4. Monotonic sequences

   • A sequence is monotonic if it is either entirely non-increasing or non-decreasing.
   • Monotonic sequences can help in establishing convergence; if a sequence is bounded and monotonic, it converges.
   • Monotonicity can be tested by comparing consecutive terms.

5. Bounded sequences

   • A sequence is bounded if there exists a real number that serves as an upper limit and a lower limit for all its terms.
   • Bounded sequences can be either convergent or divergent; however, if a bounded sequence is also monotonic, it must converge.
   • Understanding boundedness is essential for applying theorems related to convergence.

6. Arithmetic sequences

   • An arithmetic sequence is defined by a constant difference between consecutive terms, known as the common difference.
   • The general form is ( a_n = a_1 + (n-1)d ), where ( d ) is the common difference.
   • Arithmetic sequences are linear and can be easily summed using formulas.

7. Geometric sequences

   • A geometric sequence is defined by a constant ratio between consecutive terms, known as the common ratio.
   • The general form is ( a_n = a_1 \cdot r^{(n-1)} ), where ( r ) is the common ratio.
   • Geometric sequences can converge or diverge based on the value of the common ratio ( r ).

8. Recursive sequences

   • A recursive sequence is defined by a recurrence relation, where each term is defined in terms of previous terms.
   • Common examples include the Fibonacci sequence, where each term is the sum of the two preceding terms.
   • Understanding recursive sequences often requires solving the recurrence relation to find a closed form.

9. Squeeze theorem for sequences

   • The squeeze theorem states that if a sequence is "squeezed" between two other sequences that converge to the same limit, then it also converges to that limit.
   • This theorem is useful for proving the convergence of sequences that are difficult to analyze directly.
   • It is often applied in cases where the terms of the sequence can be bounded by simpler sequences.

10. Infinite series and their relationship to sequences

    • An infinite series is the sum of the terms of an infinite sequence, often expressed as ( \sum_{n=1}^{\infty} a_n ).
    • The convergence of a series is closely related to the behavior of its sequence of partial sums.
    • Understanding sequences is fundamental to studying series, as many convergence tests for series rely on the properties of the underlying sequences.