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If the sequence (n) converges to a limit L then, by definition, for any eps>0 there exists a number N such |n-L|N. However if eps=0.5 then whatever value of N we chose we find that whenever n>max{N,L}+1, |n-L|=n-L>1>eps. Proving the first statement false by contradiction.
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What does descending in algebra mean?
Descending (in a sequence) means that a the next number is "more negative" or "closer to negative infinity" or "less positive" or "further from positive infinity" or if n is a number in a sequence and n+1 is the next number then n/n+1 > 1
What is the form of the nth term of a triangular pyramidal number sequence?
1/6 n(n+1)(n+2)
What is thenext logical number 1 2 6 42 1086?
If x(n) represents the nth number is the sequence x(n+1)=x(n)*(x(n)+1) So the next number in the sequence is 1086*(1086+1)=3263442
What is the formula to find sum of n odd numbers?
The set of odd numbers is an arithmetic sequence. Let say that the sequence has n odd numbers where the first term is a1 and the last one is n. The formula to find the sum on nth terms for an arithmetic sequence is: Sn = (n/2)(a1 + an) or Sn = (n/2)[2a1 + (n - 1)d] where d is the common difference that for odd numbers is 2. Sn = (n/2)(2a1 + 2n - 2)
What is the sum of the first 22 odd numbers?
The sum of a sequence is given by sum = n/2(2a + (n-1)d) where: n = how many a = first number of sequence d = difference between terms of sequence. For the first 22 odd numbers these are: n = 22 a = 1 d = 2 → sum = 22/2(2×1 + (22 - 1)×2)) = 22² = 484 The sum of the first n odd numbers is always n²: sum = n/2(2×1 + (n-1)2) = n/2(1 + (n-1))×2 = n(n) = n²
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.
Does -1n converge?
No, -1^n does not converge as it alternates between -1 and 1 for different values of n. This oscillation prevents the sequence from approaching a specific limit.
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.
How do you find the harmonic and geometric mean for grouped data?
The geometric-harmonic mean of grouped data can be formed as a sequence defined as g(n+1) = square root(g(n)*h(n)) and h(n+1) = (2/((1/g(n)) + (1/h(n)))). Essentially, this means both sequences will converge to the mean, which is the geometric harmonic mean.
Define a monotone sequence of real numbers?
A monotone increasing sequence {r_n | n>0} is a sequence with: n>m implies r_n >= r_m A monotone decreasing sequence {r_n | n>0} is a sequence with: n>m implies r_n <= r_m A strictly monotone increasing sequence {r_n | n>0} is a sequence with: n>m implies r_n > r_m A strictly monotone decreasing sequence {r_n | n>0} is a sequence with: n>m implies r_n < r_m Theorem. All bounded monotone sequences of real numbers have a unique limit.
What is meant by converge and diverge?
Converge means coming together or meeting at a common point, while diverge means branching off or moving apart in different directions. In mathematical terms, if a sequence or series of values approaches a specific number as they progress, they are said to converge. On the other hand, if the values in a sequence or series move further apart or do not approach a specific number, they are said to diverge.
Explain how to find the terms of Fibonacci sequence?
If the Fibonacci sequence is denoted by F(n), where n is the first term in the sequence then the following equation obtains for n = 0.
Is a geometric progression a quadratic sequence?
A geometric sequence is : a•r^n while a quadratic sequence is a• n^2 + b•n + c So the answer is no, unless we are talking about an infinite sequence of zeros which strictly speaking is both a geometric and a quadratic sequence.
Does 1 over n converge?
No. ∑(1/n) diverges. It is the special infinite series known as the "harmonic series."
What is convergence in numeric math?
A sequence of numbers, xn (where n = 1, 2, 3, ...), is said to converge to a limit L if, given any positive e, however small, it is possible to find an integer k such that |xn - L| < e for all n > k.In other words, after a certain point (k) all terms of the sequence are closer to the limit (L) than any tiny number.
What is Example of bounded sequence which is not Cauchy sequence?
((-1)^n)
The next number in the sequence 1-8-27?
64. It's the sequence f(n) = n^3
