Question

Use the ratio test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is given in the following problems. State if the ratio test is inconclusive. $$ a_{n}=n^{10} / 2^{n} $$

Short answer

The series converges by the Ratio Test since the limit \( L = \frac{1}{2} < 1 \).

Step-by-step solution

Write down the Ratio Test Formula

The Ratio Test states that a series \( \sum_{n=1}^{\infty} a_{n} \) converges absolutely if \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_{n}} \right| < 1 \). If \( L > 1 \) or \( L = \infty \), the series diverges. If \( L = 1 \), the test is inconclusive.

Determine the expression for \( a_{n+1} \)

Given \( a_{n} = \frac{n^{10}}{2^{n}} \), the expression for the next term \( a_{n+1} \) is \( a_{n+1} = \frac{(n+1)^{10}}{2^{n+1}} \).

Calculate the Ratio \( \left| \frac{a_{n+1}}{a_{n}} \right| \)

Substitute \( a_{n} \) and \( a_{n+1} \) into the Ratio Test formula:\[\left| \frac{a_{n+1}}{a_{n}} \right| = \left| \frac{(n+1)^{10} \/ 2^{n+1}}{n^{10} \/ 2^{n}} \right| = \left| \frac{(n+1)^{10} \cdot 2^{n}}{n^{10} \cdot 2^{n+1}} \right| = \left| \frac{(n+1)^{10}}{n^{10} \cdot 2} \right|\]

Simplify the Ratio

Simplify the expression further:\[\left| \frac{(n+1)^{10}}{n^{10} \cdot 2} \right| = \frac{1}{2} \left( \frac{n+1}{n} \right)^{10} = \frac{1}{2} \left( 1 + \frac{1}{n} \right)^{10}\]

Find the limit of the Ratio as \( n \to \infty \)

Evaluate the limit:\[L = \lim_{n \to \infty} \frac{1}{2} \left( 1 + \frac{1}{n} \right)^{10} = \frac{1}{2} \cdot 1^{10} = \frac{1}{2}\]Thus, \( L = \frac{1}{2} < 1 \).

Conclude based on the limit

Since \( L = \frac{1}{2} < 1 \), the series \( \sum_{n=1}^{\infty} a_{n} \) converges absolutely by the Ratio Test.

Key concepts

Series Convergence

Series convergence is a concept that determines whether a given infinite series converges to a finite value. In the realm of mathematics, particularly in calculus and analysis, determining the convergence of a series is crucial for understanding its behavior. Convergence means that as you keep adding terms of the series, they approach a specific sum or value.
For example, when we examine a series such as \( \sum_{n=1}^{\infty} a_{n} \), we are interested in knowing whether the sum of terms gets closer and closer to a particular number as \( n \) increases indefinitely.

• If the terms approach a finite limit, the series is convergent.
• If they do not, the series is divergent.

The ratio test is one of the many tools that provides a clear method to assess whether a series like this converges.

Absolute Convergence

Absolute convergence refers to a stronger form of convergence for series. A series \( \sum_{n=1}^{\infty} a_{n} \) converges absolutely if the series of absolute values \( \sum_{n=1}^{\infty} |a_{n}| \) also converges. This means that even when we ignore the sign of each term, the series still converges.
This is an essential concept because:

• If a series converges absolutely, it converges unconditionally, meaning that rearranging the terms does not affect the sum.
• An absolutely convergent series is always convergent, though the reverse is not necessarily true. It's a stronger condition and provides more assurance of convergence under various transformations or manipulations.

The Ratio Test specifically checks for absolute convergence by comparing the ratio of successive terms.

Limit Evaluation

Limit evaluation is a process used extensively in calculus and analysis to determine the value that a function or sequence approaches as the input approaches some point. In the context of the ratio test, evaluating the limit of the ratio of successive terms \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_{n}} \right| \) is a crucial step.
Evaluating this limit helps us understand the long-term behavior of a sequence and thus the series.

• If \( L < 1 \), the series converges absolutely, implying it will settle down to a finite value.
• If \( L > 1 \) or \( L = \infty \), it indicates divergence, where the series grows without bound.
• If \( L = 1 \), the ratio test becomes inconclusive, and further methods are needed.

By assessing the limit as \( n \to \infty \), we gain insight into whether the series terms are diminishing fast enough to ensure convergence.

Infinite Series

An infinite series is a sum of infinitely many terms, expressed mathematically as \( \sum_{n=1}^{\infty} a_{n} \). These series can be quite complex and require careful analysis to understand their behavior.
An infinite series contrasts sharply with finite sums which stop at a certain point. With infinite series, we need to understand the concept of convergence to decide if the series closes in on a particular number.

• Infinite series are fundamental in many areas of mathematics and physics, used for approximations, solving equations, and modeling phenomena.
• Because they contain an infinite number of terms, determining convergence is not trivial and involves tests like the ratio test for guidance.

Throughout the development and study of infinite series, mathematicians like Euler and Cauchy have contributed significant insights on how such series behave and how they can be used effectively.