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A convergent sequence is an infinite sequence whose terms move ever closer to a finite limit. For any specified allowable margin of error (the absolute difference between each term and the finite limit) a term can be found, after which all succeeding terms in the sequence remain within that margin of error.
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What is a sequence that is predictable?
A deterministic sequence - as opposed to a stochastic or random sequence.
What is the name of the sequence 1 2 4 8 16 32 64 128...?
This sequence is called the doubling sequence.
What does arithmetic sequence mean?
Sequence that has addition or (subtractions*) subtraction will be +(-4)
What is the nth term for sequence 4 9 16 25?
Please note that (a) this is a sequence of square numbes, and (b) the sequence starts at 22.
A sequence in which the terms change by the same amount each time?
Arithmetic Sequence
How do you prove that the sum of a convergent sequence divided by n will converge?
You can use the comparison test. Since the convergent sequence divided by n is less that the convergent sequence, it must converge.
A convergent sequence has a LUB or a what?
JUB
Is every cauchy sequence is convergent?
Every convergent sequence is Cauchy. Every Cauchy sequence in Rk is convergent, but this is not true in general, for example within S= {x:x€R, x>0} the Cauchy sequence (1/n) has no limit in s since 0 is not a member of S.
Examples of bounded and not convergent sequence?
(0,1,0,1,...)
Any convergent sequence is a Cauchy sequence is converse true?
no converse is not true
What is a sequence which is not convergent defined as?
It could be divergent eg 1+1+1+1+... Or, it could be oscillating eg 1-1+1-1+ ... So there is no definition for a sequence that is not convergent except non-convergent.
Is the limit exists for a monotone sequence An?
If a monotone sequence An is convergent, then a limit exists for it. On the other hand, if the sequence is divergent, then a limit does not exist.
Are lexicographic preferences continuous?
Lexicographic preferences are not continuous because of the decreasing convergent sequence.
How do you know whether the given sequence is convergent?
A sequence xn is convergent, and converges to y if given any positive number d, however small, it is possible to find a value for the index k, such thatabs(xn - y) < d for all n > k.In other words, there is a value, k, such that all elements of the sequence from xk will be closer to y than an arbitrarily small value.
Prove that every convergent sequence is a Cauchy sequence?
The limits on an as n goes to infinity is aThen for some epsilon greater than 0, chose N such that for n>Nwe have |an-a| < epsilon.Now if m and n are > N we have |an-am|=|(am -a)-(an -a)|< or= |am -an | which is < or equal to 2 epsilor so the sequence is Cauchy.
Is it true when a sequence is divergent then its subsequences are divergent explain?
Not always true. Eg the divergent series 1,0,2,0,3,0,4,... has both convergent and divergent sub-sequences.
Every uniformly convergent sequence of bounded function is uniformly bounded?
The answer is yes is and only if da limit of the sequence is a bounded function.The suficiency derives directly from the definition of the uniform convergence. The necesity follows from making n tend to infinity in |fn(x)|
