MAT 021C memorization

Sequence of Partial Sums

For a series ∑(n=1 to ∞) aₙ, the sequence {sₙ} where sₙ = a₁ + a₂ + ... + aₙ. If lim sₙ = S, the series converges to S; otherwise, it diverges.

nth Term Test for Divergence

If lim aₙ ≠ 0, the series diverges. If lim aₙ = 0, the test is inconclusive.

Geometric Series

∑(n=0 to ∞) arⁿ. Converges to a/(1−r) if |r|

Telescoping Series

∑(f(n) − f(n+1)). Partial sum sₙ = f(1) − f(n+1). Converges if lim f(n+1) exists.

Integral Test

If aₙ = f(n) (positive, decreasing), then ∑aₙ and ∫f(x)dx both converge or diverge.

p-Series

∑1/nᵖ. Converges if p>1; diverges if p≤1.

Direct Comparison Test

If 0 ≤ aₙ ≤ bₙ: ∑bₙ convergent ⇒ ∑aₙ convergent; ∑aₙ divergent ⇒ ∑bₙ divergent.

Limit Comparison Test

If lim(aₙ/bₙ) = c ∈ (0,∞), ∑aₙ and ∑bₙ behave alike. If c=0 and ∑bₙ converges, ∑aₙ converges.

Absolute Convergence

If ∑|aₙ| converges, then ∑aₙ converges absolutely (and thus converges).

Ratio Test

If lim|aₙ₊₁/aₙ| < 1, absolute convergence; >1, divergence; =1, inconclusive.

Root Test

If limⁿ√|aₙ| < 1, absolute convergence; >1, divergence; =1, inconclusive.

Alternating Series Test

For ∑(−1)ⁿaₙ (aₙ > 0): Converges if aₙ decreases and lim aₙ = 0.

Alternating p-Series

∑(−1)ⁿ⁺¹/nᵖ: p>1 ⇒ absolute conv.; 0

Term

Definition

Second Derivative Test

A method to classify critical points (a,b) of f(x,y) using the discriminant D = fₓₓ(a,b)⋅fᵧᵧ(a,b) − [fₓᵧ(a,b)]².

Discriminant (D)

The value D = fₓₓ⋅fᵧᵧ − (fₓᵧ)² used in the Second Derivative Test to determine the nature of a critical point.

Local Minimum

If D > 0 and fₓₓ(a,b) > 0 at a critical point (a,b), then f has a local minimum at (a,b).

Local Maximum

If D > 0 and fₓₓ(a,b) < 0 at a critical point (a,b), then f has a local maximum at (a,b).

Saddle Point

If D < 0 at a critical point (a,b), then f has a saddle point at (a,b).

Inconclusive Case

If D = 0, the Second Derivative Test fails, and other methods must be used to classify the critical point.

Taylor Series Expansion of e^x

Taylor Series Expansion of sin x

taylor series expansion of cos x

taylor series expansion of ln (1+x)

taylor of 1/1-x

taylor of tan inverse