Hint: $\displaystyle\frac{1}{(\ln n)^{\ln n}} = \frac{1}{e^{\ln n \ln(\ln n)}}$, and if $n$ is large enough then $\ln(\ln n)>2$.dwsmith said:$\sum\limits_{n = 2}^{\infty}\frac{1}{(\ln n)^{\ln n}}$
I am trying to show that this series diverges or converges
The convergence of a series is a concept in mathematics that refers to the behavior of an infinite sequence of numbers as its terms approach a specific value or limit. It is the process of determining whether the terms of a series approach a finite limit as the number of terms increases, or if they diverge and do not have a finite limit.
The convergence of a series is determined by using various tests, such as the ratio test, the root test, or the integral test. These tests help to determine the behavior of a series and whether it converges or diverges. The choice of test depends on the specific series being analyzed and the characteristics of its terms.
Absolute convergence refers to a series where the absolute values of the terms converge, while conditional convergence refers to a series where the terms themselves converge. In other words, absolute convergence focuses on the magnitude of the terms, while conditional convergence looks at their direction. A series can have absolute convergence but not conditional convergence, and vice versa.
The concept of convergence of a series has many real-world applications, particularly in fields such as physics, engineering, finance, and computer science. It can be used to analyze the stability and predictability of systems, the convergence of algorithms and numerical methods, and the convergence of financial investments. Understanding the convergence of a series is essential for making accurate predictions and decisions in these fields.
If a series does not converge, it is said to diverge. This means that the terms of the series do not approach a finite limit, and the series may either approach infinity or oscillate between different values. A divergent series can still have some useful properties, but it cannot be summed to a specific value like a convergent series can.