الانتقال إلى المحتوى
The Lumenary
الصفوف 9–12Analysisمُخطَّط له

Sequences and series

Arithmetic and geometric sequences, summation notation, sums of finite series, recursive definitions, and an informal look at convergence of infinite geometric series.

math.analysis.sequences-series

أظهِر كل ما يلزم للوصول إلى هذا الموضوع

لم يُكتب درس هذا الموضوع بعد. المخطط أدناه جزء من خريطة المنهج ولم تجرِ مراجعته.

يعتمد على

Functions and their graphs (precalculus)Proof: direct, contrapositive, contradiction and induction

عدد المتطلبات السابقة الإلزامية (حتى الأساس): 24

ستتعلّم

  • Find formulas for the n-th term and the partial sums of arithmetic and geometric sequences, and prove them by induction.
  • Use recursive definitions and summation notation.
  • Explain why 1 + 1/2 + 1/4 + ⋯ = 2 and decide when an infinite geometric series converges.

تعريف

  • A sequence of real numbers is a function from ℕ (or from the positive integers) to ℝ.
  • An arithmetic sequence has a constant difference between terms; a geometric sequence has a constant ratio.
  • An infinite series Σ aₙ converges to S if its partial sums approach S.

مبرهنة

  • For r ≠ 1, a + ar + ⋯ + arⁿ⁻¹ = a(1 − rⁿ)/(1 − r).
  • If |r| < 1 then Σ_{k≥0} ar^k = a/(1 − r); if |r| ≥ 1 and a ≠ 0, the series diverges.
  • The harmonic series 1 + 1/2 + 1/3 + ⋯ diverges, although its terms tend to 0.

مفاهيم خاطئة شائعة

  • Thinking a series must converge if its terms tend to zero.
  • Believing an infinite sum must be infinite.
  • Forgetting that 0.999… is a geometric series whose sum is exactly 1.

يقود إلى

Calculus of one variableRecurrences and generating functions

اكتشافات اختيارية