MATH 123 Test 1

Sequence

An ordered list of numbers a₁, a₂, a₃, … that follows a rule

Sequence converges (9.1.2)

lim aₙ = L for some finite number L

Sequence diverges

lim aₙ does not exist or is ±∞

Squeezing Theorem (9.1.5)

If aₙ ≤ bₙ ≤ cₙ and both aₙ and cₙ → L, then bₙ → L

Theorem 9.1.6

If lim |aₙ| = 0, then lim aₙ = 0

n! (factorial)

n·(n−1)·…·2·1, with 0! = 1 and (n+1)! = (n+1)·n!

Strictly increasing

aₙ < aₙ₊₁ for every n

Strictly decreasing

aₙ > aₙ₊₁ for every n

Monotone

Either increasing or decreasing

Eventually (9.2.2)

The property holds after dropping finitely many terms from the start

Difference method for monotonicity

aₙ₊₁ − aₙ > 0 → increasing; aₙ₊₁ − aₙ < 0 → decreasing

Ratio method for monotonicity

For positive terms: aₙ₊₁/aₙ > 1 → increasing; < 1 → decreasing

Derivative method for monotonicity

f′(x) > 0 → increasing; f′(x) < 0 → decreasing

Theorems 9.2.3–9.2.4

Eventually increasing and bounded above → converges; eventually decreasing and bounded below → converges

Infinite series

The sum u₁ + u₂ + u₃ + … of the terms of a sequence

nth partial sum sₙ

u₁ + u₂ + … + uₙ

Series converges (9.3.2)

The partial sums sₙ approach a finite number S, called the sum of the series

Geometric series

Σ a·rᵏ⁻¹, where each term is the previous term times r

Geometric Series Theorem (9.3.3)

Converges if |r| < 1, with sum a/(1 − r); diverges if |r| ≥ 1

Telescoping series

A series whose partial sums cancel in the middle, leaving only a few terms

Harmonic series

Σ 1/k; it diverges

Divergence Test (9.4.1)

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

Theorem 9.4.2

If Σuₖ converges, then lim uₖ = 0. The converse is false

Algebraic properties of series (9.4.3)

Sums and differences of convergent series converge; a constant multiple c ≠ 0 doesn't change convergence; dropping finitely many terms doesn't change convergence

Integral Test (9.4.4) conditions

f(x) is positive, continuous, and decreasing for x ≥ 1, with f(k) = uₖ

Integral Test (9.4.4) conclusion

The improper integral of f from 1 to ∞ is finite → series converges; infinite → series diverges

p-series

Σ 1/kᵖ

p-Series Test (9.4.5)

Converges if p > 1; diverges if p ≤ 1

Comparison Test (9.5.1)

Positive terms: smaller than a convergent series → converges; bigger than a divergent series → diverges

Informal Principle 9.5.2

Constant terms in the denominator can usually be dropped without changing convergence

Informal Principle 9.5.3

In a polynomial factor, all but the leading term can usually be dropped without changing convergence

Limit Comparison Test (9.5.4)

If ρ = lim aₖ/bₖ with 0 < ρ < +∞, both series converge or both diverge

Ratio Test (9.5.5)

ρ = lim uₖ₊₁/uₖ. ρ < 1 converges; ρ > 1 or +∞ diverges; ρ = 1 inconclusive

Root Test (9.5.6)

ρ = lim (uₖ)^(1/k). ρ < 1 converges; ρ > 1 or +∞ diverges; ρ = 1 inconclusive

When to use the Ratio Test

Terms with factorials or powers like 3ᵏ

When to use the Root Test

The whole term is raised to the k-th power

Alternating series

A series whose terms alternate in sign, like Σ(−1)ᵏ⁺¹aₖ with aₖ > 0

Alternating Series Test (9.6.1)

Converges if aₖ is decreasing and lim aₖ = 0

Alternating series error bound (9.6.2)

|S − sₙ| ≤ aₙ₊₁, and S lies between any two consecutive partial sums

Absolutely convergent (9.6.3)

Σ|uₖ| converges

Conditionally convergent

Σuₖ converges but Σ|uₖ| diverges

Theorem 9.6.4

If a series converges absolutely, then it converges

Ratio Test for Absolute Convergence (9.6.5)

ρ = lim |uₖ₊₁|/|uₖ|. ρ < 1 absolutely convergent; ρ > 1 or +∞ diverges; ρ = 1 inconclusive