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Definition
A seq is monotone if it is either
-increasing (xn\-strictly increasing (xn-decreasing (xn>/xn+1)
strictly decreasing (xn>xn+1) |
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| Thm: Monotone Convergence Theorem |
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Definition
If (xn) is monotone then (xn) is bounded iff (xn) converges.
If (xn) is increasing, bdd then (xn)->sup{xn}
If (xn) is decreasing, bdd then (xn) -> inf{xn} |
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Definition
Let xn be a seq. A subseq yn is one of the form
yn=xi(n) where i:N->N is strictly increasing
Can be used in problems where we've shown xn is bdd and monotone but need to find lim. yn=xn+1 and limyn=limxn |
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| Thm: Limit of a subsequence |
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Definition
| If xn->x and yn is a subsequence, then yn->x |
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Definition
If (yn) and (zn) are subsequences of (xn) and yn->y, zn->z and y=/=z, then (xn) diverges
Example Xn=(-1)^n
Yn=X2n =1 ->1
Zn=X2n-1 =-1 ->-1 |
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| Thm: Monotone Subsequence Theorem |
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Definition
| Every sequence (Xn) contains a monotone subsequence |
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Definition
| Xk is a peak of (Xn) if Xk>/Xn for all n for n>/k |
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Definition
| Every bounded seq has a convergent subsequence |
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| Def: lim sup of Xn, lim inf of Xn |
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Definition
| limsupxn=infsn where sn = sup{Xk|k>/n}
liminfXn=supTn where Tn=inf{Xk|k>/n}
In general, liminfXn\ |
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| Thm: Convergence with liminf/sup |
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Definition
Let (Xn) be bdd. Then (Xn) converges iff liminfXn=limsupXn
If this holds, then Xn->liminfxn=limsupxn |
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| Thm: liminfs/sups of Xn and Xnk |
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Definition
| Let {Xnk} be a subseq of the bdd seq (Xn). Then liminfXn\ |
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Definition
Say (Xn) is Cauchy if for every E>0 there exists k s.t. for every n,m >/k then
|Xn-Xm| |
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Definition
| If (Xn) is Cauchy, it is bdd |
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| Thm: Cauchy and Convergence |
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Definition
| (Xn) is Cauchy iff (Xn) converges |
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Definition
A series is the sequence of partial sums.
S1=X1
S2=X1+X2=S1+X2
S3=X1+X2+X3=S2+X3
Sn=Sn-1+Xn = EXk |
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Definition
Let r be in R.
Xn=r^n
Sn=1+r+r^2+... = [1-r^(n+1)]/[1-r]
If |r|<1 then r^n->0 so series converges to 1/(1-r)
If |r|>1 then r^n diverges
If r=-1 then (Sn) is (1,0,1,0,1,...) diverges
r=/=1 |
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Definition
If EXn converges then Xn->0
If Xn-/->0 then EXn diverges
Note: Can't use this to show a series converges |
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Definition
E1/n^p
p=1 is Harmonic Series (diverges)
p>1 converges
p<1 diverges |
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Definition
Sn=1+1/2+1/3+1/4+...+1/n
Xn=1/n
Diverges |
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Definition
Let 0\Then
1)If EYn conv, then EXn does too
2)If EXn diverges, then EYn does too |
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Definition
Let Xn,Yn>0 for all n.
Assume lim(Xn/Yn) exists, equal to r.
If
1) r=/=0, EXn, EYn behave the same
2) r=0, if EYn converges, so does EXn |
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Definition
| c in R is a cluster point of A contained in R if for any Delta>0, the set VsubDelta(c)IntersectionA\{c} is nonempty |
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| Thm: Cluster pt and convergence |
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Definition
| c is a cluster point of a set A iff there is a seq (Xn) in A converging to c |
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Definition
| c is a cluster point in A iff it can be approximated by a seq in A: there exists a seq (Xn) in A\{c} s.t. Xn->c |
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Definition
| Let f:A->R and c be a cluster pt of A.
Say L is the limit of f at c if for every E>0, there exists a Delta s.t. x in VsubDelta(c)IntersectionA\{c}, then f(x) is in VsubE(c).
For x in A, c-Deltac of f(x) may exist even if f(x) is not defined at x=c
-limits are unique when they exist |
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| Thm: Sequential Criterion for Limits |
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Definition
Let A be contained in F, f:A->R, c is a cluster pt.
Then limx->c of f(x)=L iff for EVERY seq (Xn) in A\{c} that converges to c, then f(Xn)->L. |
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Definition
Let f,g: A->F, c is a cluster pt.
Then
1) limx->c [f(x)+g(x)]=limx->cf(x) + limx->c g(x)
2)limx->c[f(x)g(x)] = limx->cf(x).limx->cg(x)
3)limx->c[f(x)/g(x)]=limx->cf(x)/limx->cg(x) if g(x) =/= 0 and limx->cg(x) =/=0. |
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Definition
Let A be contained in R and f:A->R and c in A. Say f is continuous at c if either
1) c is a cluster pt of A and limx->c of f(x) exists and is equal to f(c)
2) c is not a cluster pt
By def, f is automatically continous at any isolated pt.
f is Discontinuous at c if not cont at c.
f is cont on A if cont at all c in A |
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| Thm: Equivalent Statements of cont f |
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Definition
| Let A be contained in R, f:A->R a function, c in A. Then the following are equivalent.
1)f is cont at c
2)For every E>0, there exists Delta>0 s.t. if |x-c|0, there exists Delta>0 s.t. if x is in VsubE(c)IntersectionA then f(x) is in VsubE(f(c))
<-> f(VsubDelta(c)IntersectionA) is contained in VsubE(f(c))
<->VsubDelta(c)IntersectionA contained in f^-1(VsubE(f(c)) |
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Definition
limx->c of f(x) exists, but not equal to f(c).
f(x)=[x^2)-1]/[x-1] = x+1 for x =/=1
Easy to fix this to make it continous.... |
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| Thm: Combining Cont Functions |
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Definition
| Let f,g:A->R be cont. Then so are f+g, f-g, fg, f/g if g(x) =/=0 for all x |
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Definition
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Definition
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Definition
Let f:A->R, g:B->R, f cont at c in A, g be cont at b=f(b) in B.
Then gof is cont at c. |
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Definition
| Claim: g(x)=|x| is cont at all c |
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| Thm: Continuous Functions on (closed, bounded) intervals |
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Definition
Let f:I->R be cont where I is a closed and bdd interval. Then f is bdd.
Note: Result if false if I is not closed or bdd |
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Definition
Let f:I=[a,b]->R be cont.
Then there exists a pt c in I s.t. f(c)=supf(I)
There exists some point d in I s.t. f(d)=inff(I)
Say that f attains the max on I |
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| Thm: Intermediate Value Theorem |
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Definition
| Let f:I->R be cont. Suppose a,b are in f(I) and f(a)Then there exists a pt c in I s.t. f(c)=z. |
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| Thm: Preservation of Intervals |
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Definition
If I is any interval and f:I->R is cont, then f(I) is an interval.
Warning: f(I) may not be same kind of interval as I. |
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| Def: Uniformly Continuous |
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Definition
| We say that f:A->R is uniformly cont on A if for every E>0, there exists Delta>0 s.t. if |x-y| |
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| Thm: Uniform Continuity Theorem |
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Definition
| If f is cont on [a,b], then it is uniformly cont on [a,b] |
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| Thm: Continuous Extension Theorem |
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Definition
Let f be cont on (a,b). Then f is uniform cont on (a,b) iff there exists a continuous extension g(x) on [a,b]
(If we restrict g(x) to (a,b), it is f(x)) |
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| Lemma: Uniform and Cauchy |
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Definition
| If f is uniform cont on A and (Xn) is Cauchy in A, then f(Xn) is Cauchy |
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