Ruka hadi maudhui
The Lumenary
Darasa la 9–12AnalysisImepangwa

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

Onyesha kila kinachohitajika kufikia mada hii

Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.

Inajengwa juu ya

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

Masharti ya awali ya lazima 24 (hadi chini kabisa)

Utajifunza

  • 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.

Fasili

  • 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.

Nadharia

  • 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.

Mawazo potofu ya kawaida

  • 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.

Inaelekea kwa

Calculus of one variableRecurrences and generating functions

Ugunduzi wa hiari