Σ

Series, Sequences & Differential Equations

111 cards · 🇺🇸 English

Sequences, series, convergence tests, and basic differential equations.

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Mathematical series

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  1. 1Mathematical series
    An addition of infinitely many terms, one after the other
  2. 2Notation for an infinite series
    Σ(n=1 to ∞) aₙ or a₁ + a₂ + a₃ + ...
  3. 3Partial sum
    The sum of the first n terms: Sₙ = a₁ + a₂ + ... + aₙ
  4. 4Sum of a convergent series
    The limit of the sequence of partial sums: Σaₙ = lim(n→∞) Sₙ
  5. 5Nth truncation error
    The difference between the sum of a series and its nth partial sum
  6. 6Geometric series
    A series where each term is the previous term multiplied by a constant ratio r
  7. 7General form of a geometric series
    Σ(n=0 to ∞) arⁿ where a is the initial term and r is the common ratio
  8. 8Sum of a convergent geometric series
    a/(1-r) when |r| < 1
  9. 9Partial sum formula for a geometric series
    Sₙ = a(1-rⁿ)/(1-r) if r ≠ 1, or Sₙ = na if r = 1
  10. 10Harmonic series
    Σ(n=1 to ∞) 1/n = 1 + 1/2 + 1/3 + 1/4 + ...
  11. 11Does the harmonic series converge?
    No, the harmonic series diverges
  12. 12Alternating series
    A series where terms alternate in sign
  13. 13Alternating harmonic series
    Σ(n=1 to ∞) (-1)ⁿ⁺¹/n = 1 - 1/2 + 1/3 - 1/4 + ...
  14. 14Sum of the alternating harmonic series
    ln(2)
  15. 15Telescoping series
    A series where consecutive terms cancel out: Σ(bₙ - bₙ₊₁)
  16. 16Nth-term test (vanishing condition)
    If lim aₙ ≠ 0, then Σaₙ diverges; if lim aₙ = 0, the test is inconclusive
  17. 17Absolute convergence
    A series Σaₙ is absolutely convergent if Σ|aₙ| converges
  18. 18Conditional convergence
    A series that converges but does not converge absolutely
  19. 19Is the alternating harmonic series absolutely or conditionally convergent?
    Conditionally convergent (it converges, but Σ1/n diverges)
  20. 20Relationship between absolute and unconditional convergence
    For real/complex series, absolute convergence if and only if unconditional convergence
  21. 21Riemann series theorem
    Any conditionally convergent series of real numbers can be rearranged to converge to any real number or diverge
  22. 22Direct comparison test
    If 0 ≤ aₙ ≤ bₙ and Σbₙ converges, then Σaₙ converges; if Σaₙ diverges, then Σbₙ diverges
  23. 23Limit comparison test
    If lim(aₙ/bₙ) = L > 0, then Σaₙ and Σbₙ either both converge or both diverge
  24. 24Ratio test
    If lim |aₙ₊₁/aₙ| < 1, the series converges absolutely; if > 1, it diverges; if = 1, inconclusive
  25. 25Root test
    If lim |aₙ|^(1/n) < 1, the series converges absolutely; if > 1, it diverges; if = 1, inconclusive
  26. 26Integral test
    If f is positive, decreasing, and continuous on [1,∞) with f(n)=aₙ, then Σaₙ and ∫f(x)dx converge or diverge together
  27. 27Alternating series test (Leibniz test)
    If aₙ > 0, aₙ is decreasing, and lim aₙ = 0, then Σ(-1)ⁿaₙ converges
  28. 28Error bound for alternating series
    If the alternating series test applies, |S - Sₙ| ≤ aₙ₊₁
  29. 29Cauchy's condensation test
    If aₙ is non-negative and decreasing, then Σaₙ and Σ2ᵏa₂ᵏ converge or diverge together
  30. 30Dirichlet's test
    If aₙ decreases to 0 and the partial sums of bₙ are bounded, then Σaₙbₙ converges

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