The Ratio Test stands as one of the powerful tools at our disposal for investigating the convergence behavior of series. By examining the ratio of consecutive terms, the Ratio Test allows us to make conclusive statements about the convergence or divergence of a series. In this article, we will explore the Ratio Test, discuss its convergence criteria, and demonstrate its applications in analyzing series.
We mention that in the French mathematical school, this test is called the d’Alembert rule. This is because it was founded by the French mathematician d’Alembert.
What is a ratio test?
Before diving into the Ratio Test, let’s briefly review the concept of series convergence. Given an infinite series represented as$\sum_{n=0}^na_n$, where $a_n$ denotes the nth term of the series, convergence refers to the behavior of the series as we add more terms. If the series approaches a finite limit as the number of terms increases, we say that the series converges. On the other hand, if the series does not approach a limit or grows indefinitely, it diverges. More prcisely, we recall that a series $\sum_{n= 0}^{+\infty}u_n$ is convergent if the partial sums sequence $(S_n)_n$ defined by $S_n=u_0+cdots+u_n$ has a limit. It is called divergent if the limit is $\pm\infty$ of the limit does not exist at all.
The Ratio Test is a convergence test that investigates the limiting behavior of the ratio of consecutive terms in a series. It provides valuable insights into the convergence behavior and determines whether a series converges or diverges.
Theorem: Assume a sequence of real numbers $(u_n)_n$ satisfies $u_n>0$ and there exists a real number $\ell$ such that $$ \lim_{n\to+\infty} \frac{u_{n+1}}{u_n}=\ell.$$ Then the following assertions hold:
- The series $\sum_{n=0}^{+\infty}u_n$ is convergent if $\ell<1$;
- it is divergent if $\ell>1$;
- finally, if $\ell=1$ we cannot conclude.
An overview of the bounded ratio test
D’Alembert was a French mathematician and philosopher, born in Paris in 1717 and died in 1783. He had a successful career at law school. However, the legal profession did not appeal to d’Alembert, so he decided to take courses in medicine. Again, only after a while does he turn to mathematics. He later became one of the greatest masters of mathematics.
Examples of application of the convergent test
We give some applications of the ration test. Here we give classical series.
Example 1: Discuss the convergence of the following series \begin{align*} \sum_{n=0}^{+\infty} \frac{1}{n!},\quad \sum_{n=0}^{+\infty} \frac{n}{2^n}.\end{align*} Solution: Let us put $u_n=\frac{1}{n!}$. Then we have \begin{align*} \frac{u_{n+1}}{u_n}=\frac{\frac{1}{(1+n)!}}{\frac{1}{n!}}=\frac{1}{n+1}.\end{align*} Clearly the limit of this ratio is $0$. Then the series $\sum_{n=0}^{+\infty} \frac{1}{n!}$ is convergent. On the other hand, select $v_n=\frac{n}{2^n}$. So that $$ \frac{v_{n+1}}{v_n}=\frac{n+1}{2^{n+1}}\times \frac{2^n}{n}=2 \frac{n+1}{n}.$$ The limit of this ratio is $2$. Thus the ratio test, the series $ \sum_{n=0}^{+\infty} \frac{n}{2^n}$ is divergent.
