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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.
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.
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.
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)
What is the nth term for the sequence 0 4 12 24 40?
To find the nth term of a sequence, we first need to identify the pattern or rule governing the sequence. In this case, the sequence appears to be increasing by 4, then 8, then 12, then 16, and so on. This pattern suggests that the nth term can be represented by the formula n^2 + n, where n is the position of the term in the sequence. So, the nth term for the given sequence is n^2 + n.
