Question

Consider the series \(\sum_{n=1}^{\infty} a_{n} .\) Let $$ \rho=\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|} $$ i. If \(0 \leq \rho<1\), then \(\sum_{n=1}^{\infty} a_{n}\) converges absolutely. ii. If \(\rho>1\) or \(\rho=\infty\), then \(\sum_{n=1}^{\infty} a_{n}\) diverges. iii. If \(\rho=1\), the test does not provide any information.

Short answer

Compute \(\rho\) and classify it: \(\rho < 1\) for absolute convergence, \(\rho > 1\) for divergence, \(\rho = 1\) is inconclusive.

Step-by-step solution

Identify Series Type

We are given an infinite series \(\sum_{n=1}^{\infty} a_n\) and want to determine the convergence or divergence of this series using the root test, which involves calculating \(\rho = \lim_{n \to \infty} \sqrt[n]{|a_n|}\).

Calculate the \\rho

To apply the root test, we need to calculate \(\rho = \lim_{n \to \infty} \sqrt[n]{|a_n|}\). This expression evaluates the behavior of the term's growth as \(n\) approaches infinity. \(\rho\) is used to analyze the convergence of the series.

Classify \\rho and Conclude

Once we find \(\rho\), we compare it with 1 to determine the convergence or divergence:- If \(0 \leq \rho < 1\), the series converges absolutely.- If \(\rho > 1\) or \(\rho=\infty\), the series diverges.- If \(\rho = 1\), the root test is inconclusive.

Key concepts

Series Convergence

When we talk about series convergence, we are interested in understanding whether an infinite sum of numbers approaches a finite limit. Specifically, a series \( \sum_{n=1}^{\infty} a_n \) can either "converge" to a particular value or "diverge", meaning it does not settle at any specific limit.
Convergence is a key property in mathematical analysis as it tells us about the stability of the sum as more and more terms are added.

• Convergent Series: These have a sum that approaches a certain finite value as the number of terms increases indefinitely.
• Divergent Series: These do not settle to a finite sum. Their terms could approach infinity, oscillate, or behave irregularly.

Determining whether a series converges is crucial as it impacts how well the series can approximate values or solve problems in sciences and engineering.

Root Test

The root test is a tool to determine if an infinite series converges or diverges. It is particularly useful when dealing with series where the terms contain powers or involve exponential behavior. This test calculates the limit \( \rho = \lim_{n \to \infty} \sqrt[n]{|a_n|} \).
Here's how the root test works:

• If \( \rho < 1 \), the series converges absolutely.
• If \( \rho > 1 \) or \( \rho = \infty \), the series diverges.
• If \( \rho = 1 \), the test provides no definitive conclusion.

When using the root test, it's vital to correctly calculate the limit and compare it to 1. This makes it a reliable method for series involving terms growing exponentially or at a polynomial rate.

Infinite Series

An infinite series is the sum of an endless sequence of numbers. Represented by \( \sum_{n=1}^{\infty} a_n \), it attempts to add up every element in that sequence.
Working with infinite series requires special techniques since the number of terms is unbounded.

• Finite Series: Stops at a particular number of terms. Easier to manage and sum.
• Infinite Series: Extends indefinitely. Requires tests or methods like the root test to determine if it holds a meaningful value.

Understanding series is essential for various fields such as calculus, where infinite series can represent functions as sums; this is fundamental in Fourier analysis, quantum mechanics, and more.

Absolute Convergence

The notion of absolute convergence refers to the behavior of a series when all of its terms are replaced by their absolute values.
A series \( \sum_{n=1}^{\infty} a_n \) is said to converge absolutely if the transformed series \( \sum_{n=1}^{\infty} |a_n| \) converges. This type of convergence is stronger than ordinary convergence as it implies convergence even when positive and negative terms are rearranged.

• Absolute Convergence: Implies that rearranging terms won't affect the convergence. This ensures stability and reliability.
• Conditional Convergence: Occurs when a series converges, but its absolute counterpart does not. Rearranging terms can affect convergence in these cases.

Absolute convergence is a key concept because it guarantees that the value of the series is robust against changes in the order of terms, making it relevant for complex mathematical and scientific computations.