Problem 57
Question
For \(k \in \mathbb{N}\), let \(a_{k} \in \mathbb{R}\) and define \(\alpha:=\lim \sup _{k \rightarrow \infty}\left|a_{k}\right|^{1 / k}\). Show that if \(\alpha<1\), then \(\sum_{k=1}^{\infty} a_{k}\) is absolutely convergent and if \(\alpha>1\), then \(\sum_{k=1}^{\infty} a_{k}\) is divergent.
Step-by-Step Solution
Verified Answer
Using the Root Test, we have shown that if \(\alpha = \limsup_{k\rightarrow\infty} |a_k|^{1/k} < 1\), then the series \(\sum_{k=1}^{\infty} a_k\) is absolutely convergent, and if \(\alpha > 1\), then the series is divergent.
1Step 1: Remembering the Root Test for series convergence
The Root Test states that for a series \(\sum_{k=1}^{\infty} a_k\), let
\[L = \limsup_{k \rightarrow \infty} |a_k|^{1/k}\]
If \(L < 1\), then the series is absolutely convergent, and if \(L > 1\), then the series is divergent.
2Step 2: Showing absolute convergence when α < 1
Given that \(\alpha = \limsup_{k\rightarrow\infty} |a_k|^{1/k}\), if \(\alpha < 1\), the Root Test tells us that the series \(\sum_{k=1}^{\infty} a_k\) is absolutely convergent. Therefore, if \(\alpha < 1\), we can conclude that the series is absolutely convergent.
3Step 3: Showing divergence when α > 1
Similarly, if \(\alpha > 1\), the Root Test tells us that the series \(\sum_{k=1}^{\infty} a_k\) is divergent. Therefore, if \(\alpha > 1\), we can conclude that the series is divergent.
In conclusion, we have shown that if \(\alpha = \limsup_{k\rightarrow\infty} |a_k|^{1/k} < 1\), then the series \(\sum_{k=1}^{\infty} a_k\) is absolutely convergent, and if \(\alpha > 1\), then the series is divergent.
Key Concepts
Root TestAbsolute ConvergenceDivergent SeriesLimit Superior
Root Test
When analyzing the behavior of an infinite series, several tests can determine whether it converges or diverges. One such method is the Root Test. The Root Test is particularly useful because it involves a simple calculation of a limit, rather than requiring multiple steps or comparisons.
The process is straightforward: you take the k-th root of the absolute value of the k-th term and then find the limit superior of this sequence as k approaches infinity. If the limit is less than one, you have a series that absolutely converges. On the other hand, if the limit is greater than one, the series will not settle to a finite value and is considered divergent. The test does not conclude anything for the case where the limit equals one; in such scenarios, we must use other convergence tests.
Understanding the application of the Root Test is a powerful tool in a student's mathematical toolkit, enabling them to handle a broad class of series and determine their convergence with a simple computation.
The process is straightforward: you take the k-th root of the absolute value of the k-th term and then find the limit superior of this sequence as k approaches infinity. If the limit is less than one, you have a series that absolutely converges. On the other hand, if the limit is greater than one, the series will not settle to a finite value and is considered divergent. The test does not conclude anything for the case where the limit equals one; in such scenarios, we must use other convergence tests.
Understanding the application of the Root Test is a powerful tool in a student's mathematical toolkit, enabling them to handle a broad class of series and determine their convergence with a simple computation.
Absolute Convergence
Absolute convergence is a stronger form of convergence for series. A series is said to converge absolutely if the sum of the absolute values of its terms converges. In other words, even if you disregard the signs (positive or negative) of each term and just sum up their absolute values, the series still approaches a finite sum.
Why is this significant? Absolute convergence indicates that the series is unconditionally convergent, meaning its terms can be rearranged without affecting the sum it converges to. This isn't always true for series that are conditionally convergent, where changing the order of terms can actually lead to different sums or no sum at all. Recognizing whether a series is absolutely convergent is crucial for certain mathematical operations, like integration or manipulation of power series.
Why is this significant? Absolute convergence indicates that the series is unconditionally convergent, meaning its terms can be rearranged without affecting the sum it converges to. This isn't always true for series that are conditionally convergent, where changing the order of terms can actually lead to different sums or no sum at all. Recognizing whether a series is absolutely convergent is crucial for certain mathematical operations, like integration or manipulation of power series.
Divergent Series
When we speak of divergent series, we refer to an infinite series that doesn’t converge to a finite limit. In such cases, the partial sums of the series do not settle down but instead continue to grow in magnitude without bound, or they oscillate without ever approaching a fixed sum.
It's essential to understand that a divergent series does not mean that the sequence of terms is heading to infinity; rather, it means that we cannot assign a finite, static sum to the series. This concept is important because it helps mathematicians differentiate between sequences that have a meaningful sum and those that do not, impacting various areas of analysis and number theory.
It's essential to understand that a divergent series does not mean that the sequence of terms is heading to infinity; rather, it means that we cannot assign a finite, static sum to the series. This concept is important because it helps mathematicians differentiate between sequences that have a meaningful sum and those that do not, impacting various areas of analysis and number theory.
Limit Superior
The notion of limit superior, or \(\limsup\), can seem daunting at first, but it's an indispensable concept in mathematical analysis. It gives us a way to describe the limiting behavior of a sequence even when a conventional limit does not exist.
Specifically, the limit superior of a sequence is the greatest limit point that the sequence can have. Intuitively, this is like finding the 'upper boundary' of what the sequence tends towards, where outlier peaks in the sequence can affect this limit, but long-term trends are what truly define it. In terms of series convergence, the limit superior provides a crucial value in the Root Test, as one can evaluate the convergence of a series even when the sequence formed by taking the k-th roots of the terms is very irregular or does not converge in a traditional sense.
Specifically, the limit superior of a sequence is the greatest limit point that the sequence can have. Intuitively, this is like finding the 'upper boundary' of what the sequence tends towards, where outlier peaks in the sequence can affect this limit, but long-term trends are what truly define it. In terms of series convergence, the limit superior provides a crucial value in the Root Test, as one can evaluate the convergence of a series even when the sequence formed by taking the k-th roots of the terms is very irregular or does not converge in a traditional sense.
Other exercises in this chapter
Problem 55
For \(k \in \mathbb{N}\), let \(a_{k} \in \mathbb{R}\) with \(a_{k}>0 .\) Show that $$ \liminf _{k \rightarrow \infty} \frac{a_{k+1}}{a_{k}} \leq \liminf _{k \r
View solution Problem 56
For \(k \in \mathbb{N}\), let \(a_{k} \in \mathbb{R}\) with \(a_{k} \neq 0 .\) If \(\left|a_{k+1}\right| /\left|a_{k}\right| \rightarrow \ell\) as \(k \rightarr
View solution Problem 58
For \(k \in \mathbb{N}\), let \(a_{k} \in \mathbb{R}\) with \(a_{k} \neq 0 .\) Define \(\alpha:=\lim \sup _{k \rightarrow \infty}\left|a_{k+1}\right| /\left|a_{
View solution Problem 59
Let \(\sum_{k=0}^{\infty} c_{k} x^{k}\) be a power series with \(c_{k} \neq 0\) for all \(k \in \mathbb{N}\) and let \(r\) denote its radius of convergence Prov
View solution