Exam 2 Flashcards
increasing sequences
an+1≥an
decreasing sequences
an+1<an
monotone sequences
A sequence is monotone if it is either increasing or decreasing
subsequence
Let (an) be a sequence of real numbers, and let n1<n2<n3… be an increasing sequence of natural numbers. Then the sequence (an1,an2,an3…) is called a subsequence of (an) and denoted by (ank)
convergence of a series
Σbn converges to B if the sequence (sm) converges to B
divergence of a series
A series diverges if its subsequence also diverges
sequence of partial sums
sm = b1 + b2 + b3 + b4+…+bm
harmonic series
Σ1/n
geometric series
of the form Σar^k = a + ar + ar^2 + ar^3…
Absolute convergence
If the series Σan converges then Σ|an| converges as well
conditional convergence
If the series Σan converges but Σ|an| does not converge then we say it converges conditionally
alternating series
Terms alternate between positive and negative
open set
a set O⊆R is open if for all the points a∈O there exists an ε-neighborhood Vε(a)⊆O
limit point
A point x is a limit point of a set A if every ε-neighborhood Vε(x) intersects the set A at some point other than x
closed set
A set F⊆R is closed if it contains its limit points
complement
Ac = {x∈R : x∉A}
closure
Given a set A⊆R, let L be the set of all limit points of A. The closure of A is defined to be Abar = A U L
compact set
A set K⊆R is compact if every sequence in K has a subsequence that converges to a limit also in K.
Closed and Bounded
perfect set
A set P⊆R is perfect if it is closed and contains no isolated points
bounded set
A set A⊆R is bounded if there exists M>0 such that |a|≤M for all a∈A
Bolzano-Weierstrass Theorem
Every bounded sequence contains a convergent subsequence.
the Algebraic limit theorem for series
If Σak = A and Σbk = B, then
i) Σcak = cA
ii)Σ(ak+bk) = A+B
Comparison test
Assume (ak) and (bk) are sequences satisfying 0≤ak≤bk
i) if Σbk converges Σak converges
ii) If Σak diverges then Σbk diverges
Absolute convergence test
If the series Σ|an| converges then Σan converges as well
alternating series test
i) a1≥a2≥a3…
ii) an → 0
then Σ(-1)^n-1*an converges
Heine-Borel theorem
Let K be a subset of R. All of the following statements are equivalent in the sense that one implies the other two:
i) K is compact
ii) K is closed and bounded
iii) Every open cover for K has a finite subcover
uncountability of perfect sets
A nonempty perfect set is uncountable
Cauchy Criterion for series.
The series Σak converges if and only if, given ε>0, there exists an N∈N such that whenever it follows n > m ≥ N it follows that |am+1, am+2, … , an| < ε
Monotone Convergence Theorem
If a sequence is monotone and bounded then it converges
