math 104 chapter 3: sequences and series

sequence

a sequence in metric space X is a function from the positive integers to X, written {pₙ} or (pₙ): p₁, p₂, p₃, …

convergence and divergence

(pₙ) in a metric space X converges to p ∈ X, written pₙ → p or lim pₙ = p, if for every ε > 0 there is an integer N such that d(pₙ, p) < ε for all n ≥ N; eventually every term is within ε of p, however small ε is

if no such p exists, the sequence diverges

properties of convergence

• pₙ → p if and only if every neighborhood of p contains all but finitely many of the pₙ

• if pₙ → p and pₙ → p’, then p = p’

• if (pₙ) converges, then it is bounded

• if p is a limit point of E, then some sequence of points of E converges to p. If p ∈ Ē, some sequence in E converges to p

algrebra of limits in ℝ and ℝᵏ

for real sequences, limits respect sums, products, and quotients; a sequence in ℝᵏ converges if and only if each coordinate sequence converges

subsequence

(p_{_}n_k) where n₁ < n₂ < n₃ < … are positive integers

subsequential limit

limit of a subsequence that converges

subsequence convergence

a sequence converges to p if and only if every subsequence converges to p

Bolzano-Weierstrass

in a compact metric space, every sequence has a convergent subsequence; in particular, every bounded sequence in ℝᵏ has a convergent subsequence

open/closed: the set of subsequential limits of a sequence in a metric space?

closed; is a nice example of closedness arising naturally

Cauchy sequence

For every ε > 0 there is an N such that d(pₙ, pₘ) < ε for all n, m ≥ N. The terms bunch up with each other, with no limit mentioned

convergence vs. Cauchy condition

convergence compares terms to a limit whereas Cauchy condition compares the terms to each other and is intrinsic

relationship between convergence and Cauchy condition

every convergent sequence is Cauchy; in a compact metric space and in ℝᵏ, every Cauchy sequence converges

complete metric space

metric space in which every Cauchy sequence converges; ℝ and ℝᵏ are complete while Q and (0, 1) are not

monotonically increasing and decreasing

monotonically increasing if sₙ ≤ sₙ₊₁ for all n, decreasing if sₙ ≥ sₙ₊₁ for all n

monotone convergence theorem

a monotone real sequence converges if and only if it is bounded; for an increasing bounded sequence, the limit is its supremum

upper and lower limits

Let E be the set of all subsequential limits of (sₙ) in the extended reals, allowing +∞ and −∞. Then lim sup sₙ = sup E and lim inf sₙ = inf E

unique properties of s* = lim sup sₙ

s* ∈ E (the upper limit is itself a subsequential limit); for every x > s* there is N with sₙ < x for all n ≥ N (nothing is above s* in the long run)

equivalent formula

a real sequence converges (in the extended sense) if and only if lim inf sₙ = lim sup sₙ, and then the common value is the limit

special sequences used as references/benchmarks for comparison

• if p > 0, then n^-p → 0 and p^1/n → 1

• n^1/m → 1

• if p > 0 and α ∈ ℝ, then n^α/((1 + p)ⁿ) → 0 (exponential beats powers)

• if |x| < 1, then xⁿ → 0

series

∑ aₙ means the sequence of partial sums sₙ = a₁ + … + aₙ; “sum” is a limit of partial sums, not an infinite addition

convergence and divergence of a series

converges if sₙ → s, otherwise it diverges

Cauchy criterion and n^th term test

the series ∑ aₙ converges if and only if for every ε > 0 there is N such that |aₙ + aₙ₊₁ + … + aₘ| ≤ ε for m ≥ n ≥ N

series of nonnegative terms

converges if and only if its partial sums are bounded

series comparison

if |aₙ| ≤ cₙ for large n and ∑ cₙ converges, then ∑ aₙ converges; if aₙ ≥ dₙ ≥ 0 for large n and ∑ dₙ diverges, then ∑ aₙ diverges — only the tail matters

geometric series

∑ₙ₌₀^∞ xⁿ = 1/(1 − x) if |x| < 1, and diverges if |x| ≥ 1

Cauchy Condensation

if a₁ ≥ a₂ ≥ … ≥ 0, then ∑ aₙ converges if and only if ∑ 2ᵏ a_{2ᵏ} converges

p series

the series ∑ 1/n^p converges if and only if p > 1

logarithmic series

the series ∑ 1/(n(log n)^p) converges if and only if p > 1

root test

Let α = lim sup |aₙ|^(1/n). If α < 1, the series converges; if α > 1, it diverges; if α = 1, the test says nothingr

ratio test

If lim sup |aₙ₊₁/aₙ| < 1, the series converges. If |aₙ₊₁/aₙ| ≥ 1 for all large n, it diverges

root test vs. ratio test

root test is strictly stronger; whenever the ratio test proves convergence, the root test does too (divergence branches are not comparable in this way)

power series

∑ cₙzⁿ. Let α = lim sup |cₙ|^(1/n) and R = 1/α; the series converges if |z| < R and diverges if |z| > R — R is the radius of convergence

summation by parts and the Dirichlet-type test

If the partial sums of ∑ aₙ are bounded, b₁ ≥ b₂ ≥ … and bₙ → 0, then ∑ aₙbₙ converges. Special case: the alternating series test (terms alternate in sign, decrease in absolute value, and tend to 0)

rearrangements

A rearrangement lists the same terms in a different order. If ∑ aₙ converges absolutely, every rearrangement converges to the same sum

Riemann’s Rearrangement Theorem

If a real series converges nonabsolutely, it can be rearranged to converge to any real number