Analysis

Question 1

Strictly increasing

Answer 1

a_n is strictly increasing if, for all n, a_n < a_(n+1)

Question 2

Increasing

Answer 2

a_n is increasing if, for all n, a_n ≤ a_(n+1)

Question 3

Strictly decreasing

Answer 3

a_n is strictly decreasing if, for all n, a_n > a_(n+1)

Question 4

Decreasing

Answer 4

a_n is decreasing if, for all n, a_n ≥ a_(n+1)

Question 5

Monotonic

Answer 5

a_n is monotonic if it is increasing or decreasing or both

Question 6

Non-monotonic

Answer 6

a_n is non-monotonic if it is neither increasing or decreasing

Question 7

Bounded above

Answer 7

a_n is bounded above if there exists U such that, for all n, a_n ≤ U

Question 8

Upper bound

Answer 8

U is an upper bound for a_n if for all n, a_n ≤ U

Question 9

Bounded below

Answer 9

a_n is bounded below if there exists L such that, for all n, a_n ≥ L

Question 10

Lower bound

Answer 10

L is an upper bound for a_n if for all n, a_n ≥ L

Question 11

Bounded

Answer 11

a_n is bounded if it is both bounded above and below

Question 12

Tends to infinity

Answer 12

a_n tends to infinity if, for every C>0, there exists n ∈ ℕ such that a_n>C for all n>N

Question 13

Tends to zero

Answer 13

a_n tends to zero if, for each ε>0, there exists N ∈ ℕ such that |a_n|N

Question 14

Tends to a

Answer 14

a_n tends to a if, for each ε>0, there exists N ∈ ℕ such that |a_n -a|N

Question 15

Eventually

Answer 15

a_n satisfies a certain property eventually if there is N ∈ ℕ such that the sequence a_(N+n) satisfies that properly

Question 16

Subsequence

Answer 16

A subsequence of a_n is a sequence of the form a_n_i, where (n_i) is a strictly increasing sequence of positive numbers

Question 17

Rational number

Answer 17

A real number is rational if it can be written in the form p/q, where p and q are integers with q ≠ 0

Question 18

Irrational number

Answer 18

A real number that is not rational

Question 19

Least upper bound

Answer 19

A number u is a least upper bound of A if

1. u is an upper bound of A and
2. if U any upper bound of A then u ≤ U

Question 20

Greatest lower bound

Answer 20

A number l is a greatest lower bound of A if

1. l is a lower bound of A and
2. if L is any lower bound of A then l ≥ L

Question 21

Series

Answer 21

A series is an expression of the form Σ (n=1, ∞) a_n = a_1 + a_2 + a_3 + …

Question 22

Partial sum

Answer 22

A series’ partial sum (s_n) takes the form s_n = a_1 + a_2 + … + a_n = Σ (i=1, n) a_i

Question 23

Converges (series)

Answer 23

Σ (n=1, ∞) a_n converges if (s_n) converges

Question 24

Sum of the series

Answer 24

If s_n → S then we call S the sum of the series and we write Σ (n=1, ∞) a_n = S

Question 25

Diverges (series)

Answer 25

Σ (n=1, ∞) a_n diverges if (s_n) does not converge

Question 26

Diverges to infinity (series)

Answer 26

Σ (n=1, ∞) a_n diverges if (s_n) tends to infinity

Question 27

e

Answer 27

Σ (n=1, ∞) 1/(n-1)! = 1 + 1 + 1/2! + 1/3! + …

Question 28

Absolutely convergent

Answer 28

The series Σa_n is absolutely convergent if Σ|a_n| converges

Question 29

Rearrangement

Answer 29

The sequence (b_n) is a rearrangement of (a_n) if there exists a bijection σ: ℕ → ℕ (ie a permutation on ℕ) such that b_n = a_σ(n) for all n

Question 30

Conditionally convergent

Answer 30

The series Σa_n is said to be conditionally convergent if Σa_n is convergent but Σ|a_n| is not

Question 31

Completeness axiom (upper bounds)

Answer 31

Any non-empty subset of ℝ that is bounded above has a least upper bound

Question 32

Bolzano-Weierstrauss Theorem

Answer 32

Any bounded sequence of real numbers has a convergent subsequence

Question 33

Cauchy

Answer 33

A sequence a_n is Cauchy if for every ε > 0,there exists N ∈ ℕ such that |a_n - a_m| < ε for all n,m ≥ N

Question 34

Comparison test

Answer 34

If 0 ≤ |a_n| ≤ b_n and Σb_n converges then Σa_n converges absolutely