Sequences, Series, and Limits Flashcards

Grade 11 · Math · Flashcards

Key formulas, definitions, and concepts for sequences, series, and limits.

Flashcards

Arithmetic sequence: A sequence with constant difference d between consecutive terms; a_n = a_1 + (n-1)d
Geometric sequence: A sequence with constant ratio r between consecutive terms; a_n = a_1 * r^(n-1)
Arithmetic series sum (S_n): S_n = n/2 * (a_1 + a_n) or n/2 * [2a_1 + (n-1)d]
Geometric series sum (finite): S_n = a_1(1 - r^n)/(1 - r), where r ≠ 1
Infinite geometric series sum: S = a_1/(1 - r), valid only when |r| < 1
Convergent sequence: A sequence where lim(n→∞) a_n = L for some finite L
Divergent sequence: A sequence that does not approach any finite limit
Sigma notation Σ: Compact notation for sums; Σ_{k=1}^{n} a_k means a_1 + a_2 + ... + a_n
Limit laws: If lim a_n = L and lim b_n = M, then lim(a_n + b_n) = L + M and lim(a_n * b_n) = L*M
Squeeze theorem: If a_n ≤ b_n ≤ c_n and lim a_n = lim c_n = L, then lim b_n = L
Geometric mean: The geometric mean of consecutive terms in a geometric sequence equals the middle term
Telescoping series: A series where most terms cancel; sum often simplifies to just first and last partial terms
Ratio test (informal): For a geometric series, the series converges if |r| < 1 and diverges if |r| ≥ 1