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Definition
| The sequence {xn} is said to converge, or to be convergent, if there is a real number x so that for every ϵ>0, there is a corresponding natural number Nϵ so that for all n > Nϵ, the absolute value of xn - n < ϵ. In this case, x is called the limit of the sequence, and we write this as lim as n goes to infinity of xn = x. A sequence that does not converge is said to diverge. |
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| Axiom 3 (The Least Upper Bound Axiom) |
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Definition
| Every non-empty set with an upper bound has a least upper bound. Sometimes called the Completeness axiom. |
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Axiom 1
(Natural Numbers...) |
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Definition
| There are an infinite number of natural numbers. |
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Axiom 2
Archimedean Axiom - Form 2 |
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Definition
| For every real number a > 0, there is sone n ϵ N so that 0 < 1/n < a. |
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Definition
| Between any two positive real numbers, there must be a rational number. |
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Axiom 5
(The Bolzano-Weierstrass theorem for sequences)
If a sequence is bounded, it may not converge, but... |
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Definition
| If a sequence is bounded, it may not converge, but it must have at least one subsequence that converges, and hence at least one limit point. |
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| Axiom 6 (Nested Interval Property) |
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Definition
| If we have a sequence of closed intervals I[n] = [an, bn] so that I[n+1] is within I[n] for all n and lim as n goes to infinity of (bn-an) = 0, then there is exactly one real number x that is in all the I[n] intervals. |
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| Axiom 7 (Monotone and Bounded Theorem) |
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Definition
| If a sequence is increasing and bounded above, then it must converge. |
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Axiom 8
(Greatest Lower Bound Axiom) |
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Definition
| If a set S has a lower bound, then S must have a greatest lower bound. |
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Axiom 9
Archimedean Axiom - Form 1 |
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Definition
| If 0 < a < b, then there is some n ϵ N so that na > b. |
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| What is the notation for putting all the elements of a sequence into a Set? |
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Definition
| S = {xn: n is the natural Numbers) |
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Lemma 1.7 (Lemma 4.3)
If xn converges to x =/0, then... |
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Definition
If xn converges to x =/ 0, then there is some natural number m so that for all n > m, xn > x/2, or 1/xn < 2/x.
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Definition 1.8
The sequence xn diverges to infinity if... |
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Definition
| The sequence xn diverges to infinity if for every m > 0, there is some nm so that xn > m for all n > nm. |
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| Definition 2.1 (Cauchy Sequence) |
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Definition
| xn is called a Cauchy Sequence if for every epsilon greater than 0, there is some N[epsilon] so that for all (n,m) > N[epsilon], xn-xm < epsilon. |
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Theorem 2.5
Every convergent sequence is also a... |
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Definition
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Definition 3.1
A set S is bounded above if...
A set S is bounded below if...
A set S is bounded if... |
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Definition
there is a B s.t. s <B for all s.
there is a B s.t. s > B for all s.
it is bounded above and below. |
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Thereom 3.1 (Lemma 5.2)
u is the least upper bound for the set S if and only if... |
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Definition
u is an upper bound for S and for every epsilon > 0, there is some s[epsilon] element of S so that
u-epsilon < s[epsilon] < u. |
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Definition 3.5
A sequence of closed intervals, denoted by {In = [an,bn]}, is called a nest of closed intervals if... |
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Definition
| I[n+1] is a part of I[n] for all n, and also if the limit as n goes to infinity of l(I[n]) = 0. |
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Definition 3.6
x is a cluster point for the set S if... |
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Definition
for every epsilon greater than 0, there is some s that is an element of S so that s =/ x and
x - epsilon < s < x + epsilon. |
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Theorem 3.5
If x is a cluster point for the set S, then... |
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Definition
for every epsilon greater than 0, there are infinitely many values s[n] element of S so that
x - epsilon < s[n] < x + epsilon. |
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Theorem 3.6
The Bolzano-Weierstrass theorem for sets |
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Definition
| Every bounded, infinite set has at least one cluster point. |
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Definition 4.3
x is a limit point for the sequence {xn} if...
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Definition
| there is a subsequence of {xn} that converges to x. |
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Definition 5.1
From the given sequence {xn}, form the sequence of partial sums {Sn}, where Sn = x1 + x2 + ... + xn, which equals the sum from i=1 to n of xi. If the sequence {Sn} converges to x, then... |
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Definition
| we say that the infinite series (Sigma n=1 to infinity xn) converges to x. If lim Sn does not exist as a finite value, then we say that the series (Sigma n=1 to infinity xn) diverges. |
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| lim as x goes to c of f(x) = L means |
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Definition
| For every epsilon > 0, there is a delta > 0 so that f(x) - L is < epsilon for x-c < delta |
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| lim as x goes to c f(x) = +infinity |
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Definition
For every R > 0, there is a delta > 0 so that f(x) > R for x-c < delta
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| lim as x goes to c f(x) = -infinity |
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Definition
For every R > 0, there is a delta > 0 so that f(x) < -R for x-c < delta
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| lim as x goes to infinity f(x) = L |
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Definition
For every epsilon > 0, there is a D > 0 so that f(x) - L < epsilon for x > D
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| lim as x goes to infinity f(x) = +infinity |
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Definition
For every R > 0, there is a D > 0 so that f(x) > R for x > D
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| lim as x goes to infinity f(x) = -infinity |
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Definition
| For every R > 0, there is D > 0 so that f(x) < -R for x > D |
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| lim as x goes to -infinity f(x) = L |
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Definition
For every epsilon > 0, there is D > 0 so that f(x) - L < epsilon for x < -D
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| lim as x goes to -infinity f(x) = +infinity |
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Definition
For every R > 0, there is D > 0 so that f(x) > R for x < -D
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| lim as x goes to -infinity f(x) = -infinity |
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Definition
For every R > 0, there is D > 0 so that f(x) < -R for x < -D
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| xn = 1 + r + r^2 +... converges to |
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Definition
1/(1-r) if r<1
A proof of this result is based on the sequence of partial sums concept and the cancellation principle. |
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| xn = 1 + 1/2^p + 1/3^p + ... + 1/n^p ... |
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Definition
| converges by the integral test or the limit comparison test. For p=2, it converges to pi^2/6. |
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| x[n+1] = 3 - 1/x[n] with x[1] = 1 converges... |
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Definition
| to 2 + Tao (where Tao is the value of the golden ratio). We can prove it is convergent by proving it is monotone and bounded using mathematical induction. |
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| x[n] = 1/(n+1) + 1/(n+2) + ... + 1/2n converges to... |
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Definition
| ln2. Here we prove that {xn} is monotone and bounded by use of the cancellation principle. The fact that the sequence converges to ln2 can be verified by using the standard calculus form of the definition of the integral 1 to 2 (1/t) dt. |
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| x[n+1] = .5(x[n] + 2/x[n]) with x[1] = 1 converges... |
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Definition
| to square root of 2. This is another recursively defined sequence. The fact that it is monotone and bounded can be shown directly and not by induction. |
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| x[n] = (1+(1/n))^n converges to... |
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Definition
| e. We can use the binomial theorem expression for (a+b)^n in order to prove that the series converges. |
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| xn converges to x if every interval about x contains... |
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Definition
| all but a finite number of the elements of xn |
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| x is a limit point of xn if every interval about x contains... |
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Definition
| an infinite number of elements from xn |
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| x is a cluster point for the set S if every interval about x contains... |
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Definition
| an infinite number of elements from the set. |
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| x = sup S if every interval about x contains... |
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Definition
| at least one element from the set. |
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| xn diverges to infinity if every neighborhood of infinity contains... |
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Definition
| all but a finite number of the elements of xn. |
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| xn is unbounded if every neighborhood of infinity contains... |
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Definition
| an infinite number of xn terms. |
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