Problem 2
Question
What can you say about the series \(\Sigma a_{n}\) in each of the following cases? (a) $$\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=8$$ (b) $$\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=0.8$$ (c) $$\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=1$$
Step-by-Step Solution
Verified Answer
(a) Diverges, (b) Converges, (c) Inconclusive.
1Step 1: Ratio Test Introduction
The ratio test is used to determine the convergence or divergence of infinite series by examining the limit of the absolute value of the ratio of consecutive terms. If \( L = \lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| \) is computed, we compare it to 1 to decide if the series converges or diverges.
2Step 2: Case (a) Analysis
Given \( \lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| = 8 \), which is greater than 1, the series \( \Sigma a_{n} \) is divergent. According to the ratio test, if \( L > 1 \), the series diverges.
3Step 3: Case (b) Analysis
Given \( \lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| = 0.8 \), which is less than 1, the series \( \Sigma a_{n} \) is convergent. According to the ratio test, if \( L < 1 \), the series converges.
4Step 4: Case (c) Analysis
Given \( \lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| = 1 \), the ratio test is inconclusive. If \( L = 1 \), the ratio test does not provide information about convergence or divergence, so further analysis of \( \Sigma a_{n} \) is needed using other tests or methods.
Key Concepts
ConvergenceDivergenceInfinite Series
Convergence
When we talk about convergence concerning infinite series, we're looking at whether summing an infinite number of terms leads to a finite number. In simpler terms, does adding up an endless number of terms eventually stabilize into a particular number? This property is crucial in many areas of mathematics and applied sciences.
The ratio test helps us determine convergence by examining the ratio of consecutive terms of a series. To apply this, we calculate \( L = \lim_{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| \). If this limit \( L \) is less than 1, it indicates that the terms of the series are getting smaller rapidly enough to ensure the series converges.
For instance, if \( L = 0.8 \), as in case (b), the series is converging. This means that as we add more terms, they contribute less and less to the sum, leading to stability eventually. Convergence is key in ensuring the calculations remain reliable in practical applications. With a convergent series, one can compute the sum to a desired precision by taking sufficiently many terms.
The ratio test helps us determine convergence by examining the ratio of consecutive terms of a series. To apply this, we calculate \( L = \lim_{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| \). If this limit \( L \) is less than 1, it indicates that the terms of the series are getting smaller rapidly enough to ensure the series converges.
For instance, if \( L = 0.8 \), as in case (b), the series is converging. This means that as we add more terms, they contribute less and less to the sum, leading to stability eventually. Convergence is key in ensuring the calculations remain reliable in practical applications. With a convergent series, one can compute the sum to a desired precision by taking sufficiently many terms.
Divergence
Divergence in the context of infinite series occurs when summing the terms leads to either growing indefinitely or failing to stabilize to a finite number. Essentially, the series keeps increasing as more terms are added without approaching a particular value.
The ratio test can show divergence when the limit \( L \) of the ratio of consecutive terms of the series exceeds 1. If \( L > 1 \), the terms are not shrinking fast enough and thus the series is divergent.
For example, in case (a), where \( \lim_{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| = 8 \), the series is divergent because 8 is greater than 1. This tells us that the partial sums of the series continue to grow and will not settle towards any finite number.
Divergent series often convey that the process or phenomenon being examined cannot be contained within finite bounds, which can sometimes indicate instability or growth unchecked by any limiting factors.
The ratio test can show divergence when the limit \( L \) of the ratio of consecutive terms of the series exceeds 1. If \( L > 1 \), the terms are not shrinking fast enough and thus the series is divergent.
For example, in case (a), where \( \lim_{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| = 8 \), the series is divergent because 8 is greater than 1. This tells us that the partial sums of the series continue to grow and will not settle towards any finite number.
Divergent series often convey that the process or phenomenon being examined cannot be contained within finite bounds, which can sometimes indicate instability or growth unchecked by any limiting factors.
Infinite Series
An infinite series is a sum of an infinite sequence of terms. You can imagine it as an endlessly long list of numbers all added together. Whether this sum approaches a finite number or not defines convergence and divergence.
Infinite series can be fascinating as they extend the concept of simple addition into the realm of the infinite. They're represented as \( \Sigma a_n \), where \( a_n \) are the terms being added.
When assessing infinite series, tools like the ratio test become essential. For cases where \( \lim_{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| = 1\), the test gives no clear answer, meaning the series' behavior is more complex and requires additional evaluation. This is the scenario seen in case (c). Here, further testing may involve root tests or comparing the series to known series with established convergence or divergence patterns.
In mathematics, understanding the behavior of infinite series is crucial for analysis in fields such as engineering, physics, and computer science, where calculations often depend on the interaction of infinitely many small effects.
Infinite series can be fascinating as they extend the concept of simple addition into the realm of the infinite. They're represented as \( \Sigma a_n \), where \( a_n \) are the terms being added.
When assessing infinite series, tools like the ratio test become essential. For cases where \( \lim_{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| = 1\), the test gives no clear answer, meaning the series' behavior is more complex and requires additional evaluation. This is the scenario seen in case (c). Here, further testing may involve root tests or comparing the series to known series with established convergence or divergence patterns.
In mathematics, understanding the behavior of infinite series is crucial for analysis in fields such as engineering, physics, and computer science, where calculations often depend on the interaction of infinitely many small effects.
Other exercises in this chapter
Problem 2
Explain what it means to say that \(\Sigma_{n=1}^{\infty} a_{n}=5\)
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Suppose \(f\) is a continuous positive decreasing function for \(x \geqslant 1\) and \(a_{n}=f(n) .\) By drawing a picture, rank the following three quantities
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(a) What is the radius of convergence of a power series? How do you find it? (b) What is the interval of convergence of a power series? How do you find it?
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