Essential Calculus Formulas and Convergence Theorems

Limit Theorems and Convergence

• Root Test: lim ⁿ√a_n = L
• Ratio Test: lim a_n/a_n-1 = L
• Stolz Theorem: lim a_n/b_n = lim (a_n-a_n-1)/(b_n-b_n-1)
• Arithmetic Mean: lim (a₁ +...+ a_n)/n = lim a_n
• Geometric Mean: lim ⁿ√(a₁...a_n) = lim a_n

Convergence Criteria

• Raabe: lim n(a_n/a_n+1 - 1) > 1 (Convergent)
• Pringsheim: lim n^αa_n (if α > 1, Convergent)
• Root Test (n-th): lim ⁿ√a_n < 1 (Convergent)
• Logarithmic: lim log(1/a_n) / log(n) > 1 (Convergent)
• Absolute Convergence: If ∑|(-1)ⁿa_n| converges, then ∑(-1)ⁿa_n converges.
• Leibniz Criterion: 1) lim a_n = 0; 2) a_n+1 - a_n < 0

Series Types

• Hypergeometric Series: a_n+1/a_n = (an + b)/(n + c). Convergent if (c-b)/a > 1.
• Telescoping Series: ∑a_n = x_n - x_n+1. Sum = x₁ - lim x_n+1.

Asymptotic Equivalences

sin(a_n) ≈ a_n; tan(a_n) ≈ a_n; arcsin(a_n) ≈ a_n; arctan(a_n) ≈ a_n; 1-cos(a_n) ≈ (a_n²)/2; ln(1+a_n) ≈ a_n; e^a_n-1 ≈ a_n; (1+a_n)^k-1 ≈ ka_n

Fundamental Theorems

• Rolle: If f(x) is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), there exists c ∈ (a, b) such that f'(c) = 0.
• Lagrange: If f(x) is continuous on [a, b] and differentiable on (a, b), there exists c ∈ (a, b) such that f'(c) = (f(b)-f(a))/(b-a).
• Bolzano: If f(x) is continuous on [a, b] and f(a) * f(b) < 0, there exists c ∈ (a, b) such that f(c) = 0.

Derivatives

• tan(x) = sec²x = 1/cos²x = 1 + tan²x
• cot(x) = -csc²x = -(1 + cot²x)
• arcsin(x) = 1/√(1-x²)
• arctan(x) = 1/(1+x²)
• sinh(x) = cosh(x) = (e^x - e^-x)/2
• cosh(x) = sinh(x) = (e^x + e^-x)/2
• a^f(x) = a^f(x) * ln(a) * f'(x)

Integrals

• ∫ sec²x dx = tan(x)
• ∫ csc²x dx = -cot(x)
• ∫ sec(x)tan(x) dx = sec(x)
• ∫ csc(x)cot(x) dx = -csc(x)
• ∫ 1/√(1-x²) dx = arcsin(x)
• ∫ 1/(1+x²) dx = arctan(x)

Volumes and Areas of Revolution

• Volume OX: π∫f(x)² dx
• Volume OY: 2π∫xf(x) dx
• Area OX: 2π∫f(x)√(1 + f'(x)²) dx
• Area OY: 2π∫x√(1 + f'(x)²) dx