跳到正文
The Lumenary
九至十二年级Analysis计划中

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

显示到达此主题所需的全部内容

本主题的课程尚未编写。下面的提纲是课程地图的一部分，尚未经过审校。

前置知识

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

24 项必需的前置知识（一直追溯到底）

你将学会

  • 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.

定义

  • 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.

定理

  • 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.

常见误解

  • 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.

通向

Calculus of one variableRecurrences and generating functions

可选发现