sequences

sequences

1. Sequences:

   • A sequence is an ordered list of numbers, where each number is associated with a natural index. Formally, a sequence is a function whose domain is the set of natural numbers, and its codomain is the set of real numbers.
     • Convergence:
       • A sequence { 𝑎 𝑛 } {a n ​ } converges to a limit 𝐿 L if, for every positive real number 𝜀 ε, there exists a natural number 𝑁 N This means the terms of the sequence get arbitrarily close to 𝐿 L as 𝑛 n increases. This means the terms of the sequence get arbitrarily close to 𝐿 L as 𝑛 n increases.
       • Divergence:
         • A sequence is divergent if it does not converge to any finite limit. This can happen if the terms of the sequence grow without bound or if they oscillate without approaching a specific value
     • Monotonicity:
       • Increasing: A sequence is increasing if each term is greater than or equal to the previous one,
         • Decreasing: A sequence is decreasing if each term is less than or equal to the previous one,
           • Monotonic: A sequence is monotonic if it is either increasing or decreasing.

2. References:

   • Stewart, J. (2016). Calculus: Early Transcendentals (8th ed.). Cengage Learning.
     • Fehribach, J. D. (2022). Sequences and Series in Calculus. De Gruyter.

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