Exam 2

Question 1

increasing sequences

Answer 1

an+1≥an

Question 2

decreasing sequences

Answer 2

an+1<an

Question 3

monotone sequences

Answer 3

A sequence is monotone if it is either increasing or decreasing

Question 4

subsequence

Answer 4

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)

Question 5

convergence of a series

Answer 5

Σbn converges to B if the sequence (sm) converges to B

Question 6

divergence of a series

Answer 6

A series diverges if its subsequence also diverges

Question 7

sequence of partial sums

Answer 7

sm = b1 + b2 + b3 + b4+…+bm

Question 8

harmonic series

Answer 8

Σ1/n

Question 9

geometric series

Answer 9

of the form Σar^k = a + ar + ar^2 + ar^3…

Question 10

Absolute convergence

Answer 10

If the series Σan converges then Σ|an| converges as well

Question 11

conditional convergence

Answer 11

If the series Σan converges but Σ|an| does not converge then we say it converges conditionally

Question 12

alternating series

Answer 12

Terms alternate between positive and negative

Question 13

open set

Answer 13

a set O⊆R is open if for all the points a∈O there exists an ε-neighborhood Vε(a)⊆O

Question 14

limit point

Answer 14

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

Question 15

closed set

Answer 15

A set F⊆R is closed if it contains its limit points

Question 16

complement

Answer 16

Ac = {x∈R : x∉A}

Question 17

closure

Answer 17

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

Question 18

compact set

Answer 18

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

Question 19

perfect set

Answer 19

A set P⊆R is perfect if it is closed and contains no isolated points

Question 20

bounded set

Answer 20

A set A⊆R is bounded if there exists M>0 such that |a|≤M for all a∈A

Question 21

Bolzano-Weierstrass Theorem

Answer 21

Every bounded sequence contains a convergent subsequence.

Question 22

the Algebraic limit theorem for series

Answer 22

If Σak = A and Σbk = B, then
i) Σcak = cA
ii)Σ(ak+bk) = A+B

Question 23

Comparison test

Answer 23

Assume (ak) and (bk) are sequences satisfying 0≤ak≤bk
i) if Σbk converges Σak converges
ii) If Σak diverges then Σbk diverges

Question 24

Absolute convergence test

Answer 24

If the series Σ|an| converges then Σan converges as well

Question 25

alternating series test

Answer 25

i) a1≥a2≥a3…
ii) an → 0
then Σ(-1)^n-1*an converges

Question 26

Heine-Borel theorem

Answer 26

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

Question 27

uncountability of perfect sets

Answer 27

A nonempty perfect set is uncountable

Question 28

Cauchy Criterion for series.

Answer 28

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

Question 29

Monotone Convergence Theorem

Answer 29

If a sequence is monotone and bounded then it converges