Analysis

Question 1

Order on a set

Answer 1

An order on a set X is a relation < satisfying:
* Trichotomy: for all a,b in X, exactly one holds:
* a < b
* a = b
* b < a
* Transitivity: for all a,b,c in X, if a < b and b < c, then a < c

Question 2

Triangle inequalities

Answer 2

|x + y| <= |x| + |y|
|x - y| >= | |x| - |y| |

Question 3

Archimedean property (AP)

Answer 3

Given any x,y in R, with x,y > 0,
then there exists n0 in N s.t. n0 . x > y

(0 subscript)

Question 4

Uniqueness lemma

Answer 4

If a set S has a supremum / infimum, then it is unique

Question 5

sup ( A U B )

(A union B)

Answer 5

max { sup(A) , sup(B) }

Question 6

Definition:

Indexing set

Answer 6

I = { 0, 1, … }
where the terms are usually denoted a/n

(n subscript)

Question 7

Definition:

(a/n), n in I, a sequence in R, converges to L if…

(n subscript)

Answer 7

for all eps. > 0, exists n/0 in I s.t.:
|a/n - L| < eps. for all n in I with n > n/0

Question 8

Definition:

A sequence is bounded if…

Answer 8

The sequence is both bounded above and bounded below

Question 9

Squeeze rule

Answer 9

For (a/n), (b/n), (c/n) sequences,
if a/n <= b/n <= c/n for all n in N,
and if (a/n) and (c/n) both converge to L,
then (b/n) converges to L

Question 10

Increasing / Decreasing sequence

Answer 10

Increasing if a/n < a/n+1
Decreasing if a/n > a/n+1

(or <= / >= if not stated as strictly inc./dec.)

Question 11

Non-increasing / Non-decreasing sequence

Answer 11

Non-increasing if a/n >= a/n+1
Non-decreasing if a/n <= a/n+1

Question 12

Completeness axioms

Answer 12

• Every bounded, monotonic sequence in R converges
• Every set which is bounded above has a supremum
  (and similarly for an infimum)

Question 13

Monotonic sequence

Answer 13

A sequence with an order, either:
* (monotonically) increasing
* (monotonically) non-decreasing
* (monotonically) decreasing
* (monotonically) non-increasing

Question 14

Cauchy criterian

Answer 14

A sequence (a/n), n in I, is a Cauchy sequence if:
for every eps. > 0, exists n0 in I s.t.
|a/p - a/q| < eps. for all p,q in I, p,q > n0
equivalently
If (a/n) converges, it is a Cauchy sequence

Question 15

Inequalities of sequences

Answer 15

for (a/n), (b/n), n in I, be convergent sequences,
where a/n -> a and b/n -> b,
then if a/n <= b/n, for all n in I, then a <= b

Question 16

Bernoulli inequality

Answer 16

for all a >= 0, and n in N,
(1 + a )^n >= 1 + n.a

Question 17

Binomial theorem

Answer 17

(1 + a )^n = sum(k=0 to n) nCk . a^k

Question 18

Bolzano-Weierstrass theorem

Answer 18

for (x/n) a sequence in R, with x/n in [a,b],
then there is a subsequence (x/n/k) s.t.:
(x/n/k) converges to a limit in [a,b] as k -> inf.

Question 19

Theorem:

If (a/n) is a Cauchy sequence, then…

Answer 19

(a/n) converges
(a/n) is bounded

Question 20

Continuous and Uniformly continuous relationship

Answer 20

for a function f: [a,b] -> R:
f is continuous <=> f is uniformly continuous

Question 21

Maximum Value Theorem (MVT)

Answer 21

for f: [a,b] -> R, a continuous function,
then there exists an x(max) in [a,b] s.t.:
f(x(max)) >= f(x) for all x in [a,b]

Question 22

Intermediate Value Theorem (IVT)

Answer 22

for f: [a,b] -> R, a continuous function,
and y in ( f(a) , f(b) ),
then there exists an s in (a,b) s.t. f(s) = y

Question 23

Definition:

A function f: S -> R is differentiable at a in S iff…

Answer 23

Question 24

Differentiability and Continuity relationship

Answer 24

If a function f: S -> R is differentiable at a in S, then f in continuous at a in S

Differentiability => Continuity

Question 25

Rolle's Theorem

Answer 25

for `f: [a,b] -> R`, cotinuous on `[a,b]` and differentiable on `(a,b)`, with `f(a) = f(b)`, then there exists a `c` in `(a,b)` s.t. `f'(c) = 0`

Question 26

Mean Value Theorem (MVT)

Answer 26

for `f: [a,b] -> R`, cotinuous on `[a,b]` and differentiable on `(a,b)`, there exists a `c` in `(a,b)` s.t. `f'(c) = f(b) - f(a) / b - a`

Question 27

Cauchy Mean Value Theorem

Answer 27

for `f, g: [a,b] -> R`, continuous on `[a,b]` and differentiable on `(a,b)`, with `g'(x) != 0` for all `x` in `(a,b)`, then there exists a `c` in `(a,b)` s.t. `f'(c)/g'(c) = f(b) - f(a) / g(b) - g(a)`

Question 28

Indeterminate forms for L'Hospital's rule

Answer 28

`(0/0)` `(inf/inf)`

Question 29

Partition of `[a,b]`

Answer 29

A partition `pi` of an interval `[a,b]` is a finite sequence of points where `a = x0 < x1 < ... < xm = b`, dividing `[a,b]` into subintervals `[x/i-1,x/i]`, `i = 1,...,m`

Question 30

Norm of a partition

Answer 30

`|pi|` is the length of the longest subinterval: `delta.xi`

Question 31

Refinement of a partition

Answer 31

`pi'` is a refinement of `pi` if it is obtained from `pi` by adding points

Question 32

Upper Darboux sum

Answer 32

Question 33

Lower Darboux sum

Answer 33

Question 34

Refinement lemma

Answer 34

for `f: [a,b] -> R`, a bounded function, with `pi` and `pi'` partitions of `[a,b]`, then `L(f, pi) <= L(f, pi') <= U(f, pi') <= U(f, pi)` | `pi'` a refinement of `pi`

Question 35

Comparison lemma | (of partitions)

Answer 35

for `f: [a,b] -> R`, a bounded function, with `pi/1` and `pi/2` partitions of `[a,b]`, then `L(f, pi/2) <= U(f, pi/1)` | `pi/1` and `pi/2` any two partitions

Question 36

Upper Darboux integral

Answer 36

Question 37

Lower Darboux integral

Answer 37

Question 38

Reimann integrability conditions

Answer 38

for `f: [a,b] -> R`, a bounded function, is Reimann integrable over `[a,b]` iff either: 1. for all `eps. > 0`, there exists a partition `pi` of `[a,b]` s.t. `U(f, pi) - L(f, pi) = eps.` 4. `f` is a continuous function over `[a,b]`

Question 39

Improper integral

Answer 39

1. Interval of integration isn't a closed bound, e.g. contains an `inf.` 2. The integrand isn't continuous over the interval of integration

Question 40

Fundamental Theorem of Calculus

Answer 40

for `F: [a,b] -> R` differentiable on `(a,b)`, and `F'(x) = f(x)`, where `f(x)` is continuous, then:

Question 41

Series

Answer 41

An infinite sum of the elements of a sequence `(a/n)` from `i = 1,..,inf`

Question 42

k-th partial sum

Answer 42

`S/k`, the sum of the elements `a/k` from `n = 1,..,k`

Question 43

Convergence of a series by partial sums

Answer 43

A series converges if its sequence of partial sums, `(S/k)`, converges, where the sum to infinity of the series equals the limit of `S/k` as `n -> inf.`

Question 44

p-series

Answer 44

sum to inf. of `1/p^n` * converges if `p > 1` * diverges if `p <= 1`

Question 45

Harmonic series

Answer 45

sum to inf. of `1/n`, series diverges

Question 46

nth term test for divergence

Answer 46

If the sequence `(a/n)` diverges, or the limit of `a/n` as `n -> inf != 0`, then the series diverges

Question 47

Comparison test

Answer 47

For `0 <= a/n <= b/n` for all `n` in `N`, * if the series of `b/n` converges, then so does the series of `a/n` * if the series of `a/n` diverges, then so does the series of `b/n`

Question 48

Limit comparison test

Answer 48

for `a/n, b/n >= 0`, and limit `a/n / b/n = L`, then the series of `a/n` converges/diverges iff the series of `b/n` converges/diverges

Question 49

Ratio test

Answer 49

For a series, `a/n > 0`, if the limit `a/n+1 / a/n` = * `L < 1`, then the series converges * `L > 1`, then the series diverges * `L = 1`, then no information

Question 50

Integral test

Answer 50

for `f: [1,inf) -> R`, a decreasing, non-negative function, and `a/n = f(n)`, then the series of `a/n` converges iff the integral of `f(x)` over `[1,inf)` converges

Question 51

Alternating series test | (Leibniz criterian)

Answer 51

for a decreasing sequence of non-negative terms, if the limit as `n -> inf` of `a/n = 0`, then the series `( (-1)^n . a/n )` converges

Question 52

Converges absolutely vs Converges conditionally

Answer 52

A series converges absolutely if its series of absolute values converges, otherwise it converges conditionally

Question 53

Converges pointwise

Answer 53

A sequence of functions, `( f/n )`, converges pointwise to a function `f` if: for all x, `f/n(x)` converges to `f(x)`

Question 54

Converges uniformly

Answer 54

A sequence of functions, `( f/n )` with `f: S -> R`, converges to a function `f` uniformaly on `S` if: for all `eps. > 0`, there exists `n0` in `N` s.t. for all `n > n0`, `| f/n(x) - f(x) | < eps.`

Question 55

Answer 55

Question 56

Weierstrass M-test

Answer 56

A sequence of functions, `( f/n )` with `f: S -> R`, and `(M/n)` a sequence of real numbers which converges, s.t. `| f/n(x) | <= M/n` for all `n`, then `(f/n)` converges absolutely and uniformly on `S`

Question 57

Power series

Answer 57

A series of functions `f: S -> R`, where `f/n(x) = a/n . (x - a)^n` | centered at a

Question 58

Finding radius and interval of convergence

Answer 58

Using the Ratio test on `a/n`, * if it results in `0`, radius = `inf.` * otherwise, set `< 1` to find interval, and evaluate convergence at each end point of the interval

Question 59

Taylor series

Answer 59

Question 60

Taylor expansion of `e^x`

Answer 60

Question 61

Monotonic property of subsets

Answer 61

If A c B (a subset of), then sup(A) <= sup(B), similarly for infimum

Question 62

# Definition: Continuity of a function

Answer 62

A function `f: S -> R` is continuous at `a in S` if: for all `eps > 0`, there exists a `delta > 0` s.t. `| x - a | < delta => | f(x) - f(a) | < eps.` A function is continuous on `S` if it is continuous at all `a` in `S`.

Question 63

Limit of subsequences

Answer 63

A subsequence converges to the same limit as the sequence

Question 64

Approximation property for suprema

Answer 64

For a set of real numbers `S` with a supremum, for every `eps. > 0`, there is a point `a` in `S` s.t. `sup(S) - eps. < a <= sup(S)`

Question 65

# Definition: Uniform continuity of a function

Answer 65

A function `f: S -> R` is uniformly continuous if: for all `eps > 0`, there exists a `delta > 0` s.t. `| x - y | < delta => | f(x) - f(y) | < eps.`, for all `x, y` in `S`. A function is continuous on `S` if it is continuous at all `a` in `S`.