Question

How do you test the series \[\sum{\dfrac{1}{n\ln n}}\] from \[n\] is \[\left[ 2,\infty \right)\] for convergence \[?\]

Answer 1

Hint: To solve this type of problem, first try to understand the concept of convergent and divergent series and after that we have to understand all the test by which we can check if series is convergent or not and then apply the integral test on the given series and we will get our required answer that whether the series is convergent or not.

Complete step-by-step solution:
A series can be defined as the sum of all the terms in a sequence. However, there should be some definite relationship between the terms of the sequence. In a series, order of elements is not so important but in sequence order of elements is so important.
Types of series are as follows: Harmonic Series, Geometric Series, Alternating series and Power series.
A divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero, it diverges.
There are different methods by which we can test if the given series is convergent or not. Some methods are: Manually testing the partial sum, Comparison test, Geometric Test, Root Test and Ratio Test.
We know that the differentiation of the logarithmic function gives the reciprocal of that function. Mathematically we can represent the statement as,
\[\dfrac{d}{dx}\ln x=\dfrac{1}{x}\]
So,
\[\begin{align}
  & \dfrac{d}{dx}\ln (\ln x)=\dfrac{1}{\ln x}.\dfrac{d}{dx}\ln x \\
 & \dfrac{d}{dx}\ln (\ln x)=\dfrac{1}{\ln x}.\dfrac{1}{x} \\
 & \dfrac{d}{dx}\ln (\ln x)=\dfrac{1}{x\ln x} \\
\end{align}\]
Now integrate the above expression, we get
\[\int{\dfrac{1}{x\ln x}dx=\ln (\ln x)+C}\]
Where, \[C\] is the integration constant.
Then,
\[\underset{x\to \infty }{\mathop{\lim }}\,\ln (\ln x)=\infty \]
According to the integral test, if the value for the function is less than \[1\] then the series will converge whereas if the value for the function is greater than \[1\], then the series will diverge.
So by the integral test, we can say that the given series
\[\sum\limits_{n=2}^{\infty }{\dfrac{1}{n\ln n}}\] diverges
Hence we can conclude that the given series, instead of converging, is a diverging series.

Note: Series really plays an important role in our day to day life in many aspects such as follows: it is used to predict the financial assets, we can also perform the quantitative analysis, financial and the business analysis that helps us in important business and investment decisions.