- #1
garryh22
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Homework Statement
Does the sequence {(1/2)ln(1/n)} converge or diverge?
Homework Equations
Working it out analytically, I think it diverges. I would like to know the appropriate test to show this
The sequence {(1/2)ln(1/n)} is a mathematical sequence that is defined by taking the natural logarithm of the inverse of n (1/n) and dividing it by 2. This sequence is commonly used in calculus and can be written as (1/2)ln(1/n) = ln[(1/n)^(1/2)].
It converges. As n approaches infinity, the value of (1/n)^(1/2) approaches 0, and the natural logarithm of 0 is -∞. Therefore, the limit of the sequence is -∞, indicating that it converges.
The limit of the sequence is -∞. As n approaches infinity, the value of (1/n)^(1/2) approaches 0, and the natural logarithm of 0 is -∞.
To prove that the sequence converges, you can use the definition of convergence, which states that a sequence converges to a limit L if for any positive number ε, there exists a positive integer N such that for all n>N, the absolute value of the difference between the sequence and the limit |a_n-L| is less than ε. In this case, as n approaches infinity, the sequence approaches -∞, and for any ε, there exists a positive integer N such that for all n>N, the absolute value of (1/2)ln(1/n) - (-∞) is less than ε, proving convergence.
The sequence has significance in calculus and mathematical analysis, as it is commonly used to demonstrate the concept of convergence and divergence in sequences. It also has applications in other areas of mathematics, such as in the study of infinite series and integrals.