Exploring the root Test and determining series convergence through limiting roots. In fact, in the realm of mathematical analysis, the convergence of infinite series plays a fundamental role. Determining whether a series converges or diverges is a crucial step in understanding its behavior.
One powerful tool for investigating series convergence is the root test. By examining the limiting behavior of the nth root of the series terms, the Root Test allows us to make conclusive statements about convergence. In this article, we will delve into the root test, explore its convergence criteria, and discuss its applications in analyzing series.
Understanding Series Convergence
Before delving into the Root Test, let’s briefly review the concept of series convergence. Given an infinite series, represented as $\sum_{n=0}^\infty a_n$, 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. Conversely, if the series does not approach a limit or grows indefinitely, it diverges.
Introducing the Root Test
The Root Test is a convergence test that provides valuable insights into the behavior of a series. It focuses on the limiting behavior of the nth root of the absolute values of the series terms. The test determines convergence by evaluating the limit of the nth root as n approaches infinity.
The Convergence Criteria
Here we use series with positive terms. We will see that the root test is finer than the ratio test in the sense that if the ratio test works, the root test also works.
Theorem: Let $(u_n)_n$ be a sequence with positive terms such that there exists $L\in \mathbb{R}\cup\{+\infty\}$ such that $$ \lim_{n\to+\infty}(u_n)^{\frac{1}{n}}=L.$$ Then we have the following situations
- If $L\in[0,1)$, the series $\sum_{n=1}^{+\infty}u_n$ is convergent;
- this series is divergent in the case of $L>1$ ou $L=+\infty;$
- however, if $L=1$, we can not decide on the nature of the series.
This theorem is called the root test for series and its proof is mainly based on the comparison test for convergent series.
Proof of the theorem: Assume that $L\in [0,1)$, then there exists $a>0$ such that $L<a<1$. Now choose $\varepsilon\in (0,a-L)$. Then by the sequences limit definition, there exists an integer $N$ such that for any $n$ bigger than $N$, in particular, we have $(u_n)^{\frac{1}{n}}\le L+\varepsilon<L+a-L=a$. Thus $$ 0\le u_n\le a^n,\quad \forall n\ge N.$$ We know that the geometric series $\sum_{n=1}^{+\infty}a^n$ converges, so that the series $\sum_{n=1}^{+\infty}u_n$ is convergent.
Assume that $L\in (1,+\infty)$ and let $\delta\in (1,L)$. Choose $\varepsilon \in (0,L-\delta)$. Again by the definition of the limit of the sequence, there exists a positive integer $N$ such that for any integer $n$ bigger than $N,$ we have $L-\varepsilon<(u_n)^{\frac{1}{n}}$, so that $\delta^n\le u_n$. This is because $\delta=L-(L-\delta)<L-\varepsilon$. As $\delta>1,$ then the geometric series $\sum_{n=1}^{+\infty}\delta^n$ is divergent, then the series $\sum_{n=1}^{+\infty}u_n$ is divergent as well.
Remark: We also have the following result, the series $\sum_{n=1}u_n$ with positive terms is convergent if $\limsup_{n\to+\infty}(u_n)^{\frac{1}{n}}<1$.
This test is due to the French mathematician Augustin-Louis Cauchy, born August 21, 1789, in Paris, and died in Sceaux, on May 23, 1857.
Comparison with the ratio test
Recall that the ratio test is associated with the ratio $\frac{u_{n+1}}{u_n}$. If the limit of this ratio is less than one, the series converges and diverges if the limit is greater than one. If the limit is one, we cannot decide.
We omit the the proof of the following result $$\liminf_n \frac{u_{n+1}}{u_n}\le \limsup_{n\to+\infty}(u_n)^{\frac{1}{n}}\le \limsup_n \frac{u_{n+1}}{u_n}.$$ We deduce that if the ratio test works well, then the root test also works well for the convergence of the series.
Applications of the Root Test
Here are some more applications of the Root Test:
Analyzing Series with Exponential Terms
The Root Test is particularly useful in determining the convergence of series that involve exponential terms. For example, consider the series $\sum_{n=0}^{+\infty}a_n x^n$, where $a_n$ is a sequence of real numbers and $x$ is a real variable. By applying the Root Test to the absolute values of the terms, we can determine the convergence or divergence of the series for different values of $x$.
Investigating Series with Factorial Terms
The Root Test is applicable to series that involve factorial terms. For instance, consider the series $\sum_{n=1}^\infty \frac{a_n}{n!}$, where $a_n$ is a sequence of real numbers. By taking the nth root of the absolute values of the terms and examining the limiting behavior, the Root Test allows us to determine whether the series converges or diverges.
Let’s go through some examples to illustrate the application of the Root Test in analyzing series with exponential terms:
Example 1: Convergence of the Exponential Series $\sum^{\infty}_{n=0}\frac{1}{2^n}$. Applying the Root Test, we compute the limit as n approaches infinity of $\sqrt[n]{\frac{1}{2^n}}$. The absolute values are unnecessary in this case since the terms are positive. We have $\sqrt[n]{\frac{1}{2^n}}=\frac{1}{2}$. As n approaches infinity, this limit is equal to 1/2, which is less than 1. Therefore, the series converges absolutely.
Example 2: Convergence of the Exponential Series $\sum_{n=0}^\infty \frac{x^n}{n!}$, where n! represents the factorial of n. Applying the Root Test, we compute the limit as n approaches infinity of $\sqrt[n]{\left|\frac{x^n}{n!}\right|}=\sqrt[n]{\frac{1}{n!}}|x|$. We can simplify this limit using the properties of factorials. As n approaches infinity, the factorial term in the denominator grows rapidly, approaching infinity. Consequently, $\sqrt[n]{\frac{1}{n!}}$ approaches 0. Thus, the limit becomes 0. Since the limit is less than 1 for any value of x, the series converges absolutely.
These examples demonstrate how the Root Test can be applied to series with exponential terms to determine their convergence or divergence. By evaluating the limiting behavior of the nth root of the terms, we can make conclusions about the convergence of the series for different values of the variable x.
