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University revision flashcards > Analysis > Flashcards

Analysis Flashcards

1
Q

Order on a set

A

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

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2
Q

Triangle inequalities

A

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

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3
Q

Archimedean property (AP)

A

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

(0 subscript)

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4
Q

Uniqueness lemma

A

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

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5
Q

sup ( A U B )

(A union B)

A

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

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6
Q

Definition:

Indexing set

A

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

(n subscript)

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7
Q

Definition:

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

(n subscript)

A

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

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8
Q

Definition:

A sequence is bounded if…

A

The sequence is both bounded above and bounded below

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9
Q

Squeeze rule

A

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

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10
Q

Increasing / Decreasing sequence

A

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

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

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11
Q

Non-increasing / Non-decreasing sequence

A

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

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12
Q

Completeness axioms

A
  • Every bounded, monotonic sequence in R converges
  • Every set which is bounded above has a supremum
    (and similarly for an infimum)
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13
Q

Monotonic sequence

A

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

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14
Q

Cauchy criterian

A

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

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15
Q

Inequalities of sequences

A

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

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16
Q

Bernoulli inequality

A

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

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17
Q

Binomial theorem

A

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

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18
Q

Bolzano-Weierstrass theorem

A

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.

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19
Q

Theorem:

If (a/n) is a Cauchy sequence, then…

A

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

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20
Q

Continuous and Uniformly continuous relationship

A

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

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21
Q

Maximum Value Theorem (MVT)

A

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]

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22
Q

Intermediate Value Theorem (IVT)

A

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

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23
Q

Definition:

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

A
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24
Q

Differentiability and Continuity relationship

A

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

Differentiability => Continuity

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25
Q

Rolle’s Theorem

A

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

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26
Q

Mean Value Theorem (MVT)

A

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

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27
Q

Cauchy Mean Value Theorem

A

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)

28
Q

Indeterminate forms for L’Hospital’s rule

A

(0/0)
(inf/inf)

29
Q

Partition of [a,b]

A

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

30
Q

Norm of a partition

A

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

31
Q

Refinement of a partition

A

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

32
Q

Upper Darboux sum

A
33
Q

Lower Darboux sum

A
34
Q

Refinement lemma

A

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

35
Q

Comparison lemma

(of partitions)

A

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

36
Q

Upper Darboux integral

A
37
Q

Lower Darboux integral

A
38
Q

Reimann integrability conditions

A

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]

39
Q

Improper integral

A
  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
40
Q

Fundamental Theorem of Calculus

A

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

41
Q

Series

A

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

42
Q

k-th partial sum

A

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

43
Q

Convergence of a series by partial sums

A

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.

44
Q

p-series

A

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

45
Q

Harmonic series

A

sum to inf. of 1/n, series diverges

46
Q

nth term test for divergence

A

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

47
Q

Comparison test

A

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

48
Q

Limit comparison test

A

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

49
Q

Ratio test

A

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

50
Q

Integral test

A

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

51
Q

Alternating series test

(Leibniz criterian)

A

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

52
Q

Converges absolutely vs Converges conditionally

A

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

53
Q

Converges pointwise

A

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

54
Q

Converges uniformly

A

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.

55
Q
A
56
Q

Weierstrass M-test

A

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

57
Q

Power series

A

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

centered at a

58
Q

Finding radius and interval of convergence

A

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

59
Q

Taylor series

A
60
Q

Taylor expansion of e^x

A
61
Q

Monotonic property of subsets

A

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

62
Q

Definition:

Continuity of a function

A

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.

63
Q

Limit of subsequences

A

A subsequence converges to the same limit as the sequence

64
Q

Approximation property for suprema

A

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)

65
Q

Definition:

Uniform continuity of a function

A

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.

University revision flashcards (12 decks)

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