Analysis

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Define a field.: A non-empty set F with 2 binary operations + and . st F is an abelian group under + F* is an abelian group under . Distributivity: for all a,b,c in F a.(b+c)=(a.b)+(a.c)
Define an order.: A relation on a set X satisfying (1) Trichotomy for all a,b in X, exactly one holds: a>b, a=b, b>a (2) Transitivity: for all a,b in X c>b and b>a => c>a
Define an ordered field.: A field with an order < which is well behaved wrt + and . For all a,b,c in F (1)a < b then a+c < b+c (2)a < b then ac < bc if 0 < c
In an ordered field F, what is |xy|?: |x||y|
What is the triangle inequality?: In an ordered field |x+y|=< |x|+|y|
What is the reverse triangle inequality?: In an ordered field ||x|-|y||=< |x-y|
What does it mean for a subset A of an ordered field F to have a maximum s?: s in A for all x in A, s>=x
Under what circumstances do you know that A has a minimum and a maximum?: If A is a finite non-empty subset of an ordered field F.
What is the Archimedian property?: In an ordered field, For any x,y>0, there exists n in N st n.x>y x not infinitesimal y not infinite
What is a subsequence of a sequence an:n in I?: A sequence of the form ank:k in J where each nk lies in I and for each k in J we have nk
What does it mean if lim(n->8)an = L: for all E>0 there exists no in I st |an-L|
If a sequence an converges to L, what does the subsequence ank converge to?: L
Define the real numbers?: An ordered field satisfying: Every bounded monotonic sequence in R converges to a limit in R (a complete ordered field)
Let an be sequence converging to L and c in R. What does can converge to?: cL
Define a Cauchy sequence.: A sequence an with for all E>0 there exists an no in I st |ap-aq|
If an is not a Cauchy sequence then an is...: a diverging sequence. Converging => Cauchy.
Define [x]: The integer part or floor function. The greatest integer n st n=< x. It is well defined.
If an converges to a and bn converges to b and an=< bn for all n, what does that tell you about a and b?: a=< b
Let an be a convergent sequence in [a,b]. What can you say about lim(n->8)an?: it\'s in [a,b]
What does it tell you if an->a as n->8 and an->b as n->8?: a=b
What is the Squeeze rule?: If an=< bn=< cn for all n and if lim(n->8)an=L and lim(n->8)cn=L, then bn converges and lim(n->8)bn=L.
If c>0 and an->a, c^an->?: c^a
If an->a then cos(an)->?: cos(a), same for sin
The Bernoulli inequality: For all a>=0 and all n, (1+a)^n >= 1+na
A convergent sequence is always...: bounded.
A is bounded above by s if...: a=< s for all a in A. s is a non-unique upper bound for a.
s is the supremum of A sup(A) if...: for all x in A, s>=x if u is an upper bound for A then s=
infinum: least lower bound
If F is an ordered field where every bounded set has an infinum and a supremum...: every bounded monotone sequence in F converges in F.
Bolzano-Weierstrass Theorem: Let xn be a sequence of real numbers in [a,b]. Then there is a subsequence xnk which converges to a limit in [a,b] as k->8. [a,b] is sequentially compact.
When is a set S (subset of reals) sequentially compact?: If for every sequence xn in S there is a subsequence xnk which converges to a limit in S.
General principle of convergence: If an is a Cauchy sequence in R then an converges in R.
When does zn=xn+iyn converge to L=M+iN as n->8?: When xn->M and yn->N as n->8.
right handed limit: ...0< x-a< (d) =>...
When is f continuous at a?: lim(x->a)f(x)=f(a) ...for all x in S, |x-a|...
When is g(f(x)) cts at a?: When f is cts at a and g is cts at f(a).
When does f(an) converge to f(a)?: When an converges to a and f is continuous.
When is f uniformly continuous?: ...for all a,x in S, |x-a|< (d)... same (d) works for every a.
Let f be defined on [a,b]. When is f uniformly continuous on [a,b]?: If and only if it is continuous.
Maximum Value Theorem: Let f be cts on [a,b]. Then there is an xmax in [a,b] such that f(xmax)>=f(x) for all x in [a,b]. Similarly for minimum.
Intermediate Value Theorem: Let f be cts on [a,b] and let f(a)=< y=< f(b). Then there is an xo in (a,b) st f(xo)=y
When is f differentiable at a?: lim(h->0) (f(a+h)-f(a))/h exists
Let f(x+iy)=u(x,y)+iv(x,y). What is the partial derivative of u wrt x?: (d)u/(d)x= lim(h->0) (u(b+h,c)-u(b,c))/h
Cauchy-Riemann eqns: If f=u+iv is complex differentiable at a=b+ic then u and v have partial derivatives at (b,c) satisfying: (d)u/(d)x=(d)v/(d)y (d)u/(d)y=-(d)v/(d)x
If f is differentiable at a then f is...: continuous at a.
Rolle\'s theorem: Suppose f is continuous on [a,b] and differentiable on (a,b) and that f(a)=f(b). Then there exists a number c in (a,b) so that f\'(c)=0.
Mean value theorem: Suppose that f is continuous of [a,b] and differentiable on (a,b). Then there exists c in (a,b) so that f\'(c)=(f(b)-f(a))/(b-a)
Let f:R->R be differentiable. If f\'(x)>0 for all x, then...: f(x) is increasing.
Cauchy mean value theorem: Let f and g be continuous on [a,b] and differentiable on (a,b) and suppose that g\'(x) is non zero on (a,b). Then there exists some c in (a,b) so that (f(b)-f(a))/(g(b)-g(a)) =f\'(c)/g\'(c)
L\'Hopital\'s rule: Suppose f and g are differentiable on on I=(a-e,a)U(a,a+e) for some e>0 and that g\'(x) is non-zero on I. Suppose also that lim(x->a)f(x)= lim(x->a)g(x)=0 Then lim(x->a)f(x)/g(x)= lim(x->a)f\'(x)/g\'(x) if it exists.
When does a limit have indeterminate form?: if it has the form 0/0, 8/8 etc.
When is f:[a,b]->R a step function?: When there exist a=xo< x1< ...< xn=b such that f is constant on (xi-1,xi) for i=1,2,...,n.
How do you define the function su?: su= 1 a=< x< u 0 u=< x=< b for u in (a,b].
How can you express a step function?: For any step function f there is a unique linear combination s of finitely many su st s agrees with f except at finitely many points.
What do you know if the sum of gi.sui=0.: that all gi are 0 as sui are linearly independant.
If a linear combination of sui is 0 except possibly at finitely many points then...: itr is 0.
If f is a step function, what is I(f)?: SUM(1,n)gi.sui is the linear combination for f. Then I(f)=SUM(gi.ui)
Riemann integrable: f:[a,b}->R bounded. For every E>0 there exist functions g,h on [a,b] with g(x)=< f(x)=< h(x) for all x in [a,b] such that I(h)-I(g)< E.
Let f:(a,b)->R be Riemann integrable. What is S(a,b)f(x)dx?: sup{I(g): g step with g(x)=< f(x) for all x in [a,b]}
If f is continuous then...: f is Riemann integrable
When is an integral improper?: Interval of intergration is open or f(x) has an infinite discontinuity.
If f cont on [a,8), when does S(a,8)f(x)dx converge?: If lim(M->8)S(a,M)f(x)dx exists and is finite.
If f cont on (-8,8) when does S(-8,8)f(x)dx converge?: If for c in (-8,8) the integrals of f(x) S(c,8) and S(-8,c) both converge.
If f cont on [a,b] except at c then S(a,b)f(x)dx converges if: S(a,c) and S(c,b) both converge.
If f cont on (a,b] but infinite at a, when does S(a,b) converge?: When lim(c->a+)S(c,b) exists and is finite.
What is a series?: The sum of a sequence.
What is a sequence of partial sums?: Sk=SUM(0,k)an
If Sk is the sequence of partial sums of the series SUM(0,8)an, how do you tell if the series converges or diverges?: Same answer as the partial sum. Converges to same number.
nth term test for divergence: if an does not converge to 0 then series SUMan diverges.
Comparison test: If 0=< an=< bn for all n>=0 SUM(bn) conv => SUM(an) conv
Limit comparison test: If lim(n->8)(an/bn)=L exists with L finite then SUM(an) conv iff SUM(bn) conv
Integral test: If f(x) decreasing cont. and f(n)=an then S(0,8)f(x)dx conc iff SUM(0,8)an conv.
Ratio test: If lim(n->8)(a(n+1)/an)=L exists then L< 1 => conv L>1 => div L=1 => ?
Root test: If lim(n->8)(an)^(1/n)=L exists L< 1 => conv L>1 => div L=1 => ?
Alternating series test: If series has form SUM(0,8)((-1)^n)an where an>0 for all n: If a(n+1)=< an for all n and lim(n->8)an=0 then series converges.
If a series converges absolutely then...: it converges.
Converges conditionally: converges but doesn\'t converge absolutely
Taylor series centred at a: SUM(0,8) (f^(n)(a)/n!)(x-a)^n
Taylors theorem: Let f be (k+1) times differentiable on (a-r,a+r) and sk(x) be the taylor series of f centred at a. Then for each x in (a-r,a+r) with x=/=a there exists c between a and x such that: f(x)=sk(x)+(f^(k+1)(c)/(k+1)!)(x-a)^(k+1)

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Created By rioka