Analysis Flashcards
Order on a set
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
Triangle inequalities
|x + y| <= |x| + |y||x - y| >= | |x| - |y| |
Archimedean property (AP)
Given any x,y in R, with x,y > 0,
then there exists n0 in N s.t. n0 . x > y
(0 subscript)
Uniqueness lemma
If a set S has a supremum / infimum, then it is unique
sup ( A U B )
(A union B)
max { sup(A) , sup(B) }
Definition:
Indexing set
I = { 0, 1, … }
where the terms are usually denoted a/n
(n subscript)
Definition:
(a/n), n in I, a sequence in R, converges to L if…
(n subscript)
for all eps. > 0, exists n/0 in I s.t.:|a/n - L| < eps. for all n in I with n > n/0
Definition:
A sequence is bounded if…
The sequence is both bounded above and bounded below
Squeeze rule
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
Increasing / Decreasing sequence
Increasing if a/n < a/n+1
Decreasing if a/n > a/n+1
(or <= / >= if not stated as strictly inc./dec.)
Non-increasing / Non-decreasing sequence
Non-increasing if a/n >= a/n+1
Non-decreasing if a/n <= a/n+1
Completeness axioms
- Every bounded, monotonic sequence in R converges
- Every set which is bounded above has a supremum
(and similarly for an infimum)
Monotonic sequence
A sequence with an order, either:
* (monotonically) increasing
* (monotonically) non-decreasing
* (monotonically) decreasing
* (monotonically) non-increasing
Cauchy criterian
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
Inequalities of sequences
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
Bernoulli inequality
for all a >= 0, and n in N,(1 + a )^n >= 1 + n.a
Binomial theorem
(1 + a )^n = sum(k=0 to n) nCk . a^k
Bolzano-Weierstrass theorem
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.
Theorem:
If (a/n) is a Cauchy sequence, then…
(a/n) converges(a/n) is bounded
Continuous and Uniformly continuous relationship
for a function f: [a,b] -> R:f is continuous <=> f is uniformly continuous
Maximum Value Theorem (MVT)
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]
Intermediate Value Theorem (IVT)
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
Definition:
A function f: S -> R is differentiable at a in S iff…
Differentiability and Continuity relationship
If a function f: S -> R is differentiable at a in S, then f in continuous at a in S
Differentiability => Continuity
Rolle’s Theorem
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
Mean Value Theorem (MVT)
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
Cauchy Mean Value Theorem
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)
Indeterminate forms for L’Hospital’s rule
(0/0)(inf/inf)
Partition of [a,b]
A partition pi of an interval [a,b] is a finite sequence of points wherea = x0 < x1 < ... < xm = b,
dividing [a,b] into subintervals [x/i-1,x/i], i = 1,...,m
Norm of a partition
|pi| is the length of the longest subinterval:delta.xi
Refinement of a partition
pi' is a refinement of pi if it is obtained from pi by adding points
Upper Darboux sum
Lower Darboux sum
Refinement lemma
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
Comparison lemma
(of partitions)
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
Upper Darboux integral
Lower Darboux integral
Reimann integrability conditions
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]
Improper integral
- Interval of integration isn’t a closed bound, e.g. contains an
inf. - The integrand isn’t continuous over the interval of integration
Fundamental Theorem of Calculus
for F: [a,b] -> R differentiable on (a,b),
and F'(x) = f(x), where f(x) is continuous, then:
Series
An infinite sum of the elements of a sequence (a/n) from i = 1,..,inf
k-th partial sum
S/k, the sum of the elements a/k from n = 1,..,k
Convergence of a series by partial sums
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.
p-series
sum to inf. of 1/p^n
* converges if p > 1
* diverges if p <= 1
Harmonic series
sum to inf. of 1/n, series diverges
nth term test for divergence
If the sequence (a/n) diverges,
or the limit of a/n as n -> inf != 0,
then the series diverges
Comparison test
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
Limit comparison test
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
Ratio test
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
Integral test
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
Alternating series test
(Leibniz criterian)
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
Converges absolutely vs Converges conditionally
A series converges absolutely if its series of absolute values converges,
otherwise it converges conditionally
Converges pointwise
A sequence of functions, ( f/n ), converges pointwise to a function f if:
for all x, f/n(x) converges to f(x)
Converges uniformly
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.
Weierstrass M-test
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
Power series
A series of functions f: S -> R,
where f/n(x) = a/n . (x - a)^n
centered at a
Finding radius and interval of convergence
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
Taylor series
Taylor expansion of e^x
Monotonic property of subsets
If A c B (a subset of),
then sup(A) <= sup(B), similarly for infimum
Definition:
Continuity of a function
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.
Limit of subsequences
A subsequence converges to the same limit as the sequence
Approximation property for suprema
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)
Definition:
Uniform continuity of a function
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.
