Problem 90
Question
Prove that if \(\left\\{s_{n}\right\\}\) converges to \(L\) and \(L>0,\) then there exists a number \(N\) such that \(s_{n}>0\) for \(n>N\).
Step-by-Step Solution
Verified Answer
Yes, if a sequence \(\{s_{n}\}\) converges to a value \(L > 0\), there will exist an integer \(N\) such that for all \(n > N\), the terms of the sequence \(s_{n}\) are greater than zero. This is proven by the definition of sequence convergence and the property that a sequence converging to a positive number implies that the later terms of the sequence are also positive.
1Step 1: Understand the Definition of Sequence Convergence
Based on the definition of sequence convergence, if a sequence \(\{s_{n}\}\) converges to \(L > 0\), for any positive number \(ε\), there exists an integer \(N\) such that for all \(n > N\), the absolute value \(|s_{n} - L| < ε\). If we take \(ε\) to be equal to \(L/2\) which is a positive number as \(L > 0\), then it's evident that \(|s_{n} - L| < L/2\).
2Step 2: Analyze the Absolute Value Inequality
Given the absolute value inequality \(|s_{n} - L| < L/2\), it implies two inequalities: \(s_{n} - L < L/2\) and \(-s_{n} + L < L/2\). Simplifying these gives \(s_{n} > L/2\) and \(s_{n} > L/2\). Both of these inequalities tell us that \(s_{n} > L/2\). Therefore, we can say that all \(s_{n} > 0\) for all \(n > N\) if we select \(N\) such that this criterion is fulfilled.
3Step 3: Conclusion
Therefore, we have proved that there exists a number \(N\) such that all terms of the sequence \(s_{n}\) greater than \(N\) are also greater than zero. This conforms to the definitions of sequence convergence and provides a important property of sequences that converge to a number \(L\) which is greater than 0.
Key Concepts
Convergence of Sequences in CalculusAbsolute Value InequalitiesLimit of a Sequence
Convergence of Sequences in Calculus
Understanding the convergence of sequences is central to many concepts in calculus and analysis.
At the heart of this concept is the idea of a sequence \((s_n)\) getting 'arbitrarily close' to a limiting value \((L)\) as \((n)\) increases. To say that \((s_n)\) converges to \((L)\) means that as we look at terms \((s_n)\) with larger and larger indices, they become indistinguishable from \((L)\) within any margin of error that we choose, no matter how small. This 'margin of error' is frequently denoted by \((ε)\), a positive number that represents the radius within which \((s_n)\) must fall to be considered close to \((L)\).
The formal definition says: For every \((ε>0)\), there exists an \((N)\) such that for all \((n>N)\), \(|s_n - L| < ε\). This definition provides the backbone for proving many properties about sequences, including the exercise at hand, where it's shown that if a sequence converges to a positive number, there comes a point after which all terms of the sequence are positive.
At the heart of this concept is the idea of a sequence \((s_n)\) getting 'arbitrarily close' to a limiting value \((L)\) as \((n)\) increases. To say that \((s_n)\) converges to \((L)\) means that as we look at terms \((s_n)\) with larger and larger indices, they become indistinguishable from \((L)\) within any margin of error that we choose, no matter how small. This 'margin of error' is frequently denoted by \((ε)\), a positive number that represents the radius within which \((s_n)\) must fall to be considered close to \((L)\).
The formal definition says: For every \((ε>0)\), there exists an \((N)\) such that for all \((n>N)\), \(|s_n - L| < ε\). This definition provides the backbone for proving many properties about sequences, including the exercise at hand, where it's shown that if a sequence converges to a positive number, there comes a point after which all terms of the sequence are positive.
Absolute Value Inequalities
Dealing with absolute values often leads to discussing inequalities, since it represents the distance from zero on the number line, regardless of direction.
An inequality involving an absolute value, such as \(|s_n - L| < ε\), typically breaks down into two separate scenarios, capturing the 'distance' concept. It means the expression inside the absolute value can be less than \((ε)\) units above \((L)\) or less than \((ε)\) units below \((L)\).
In our case, by solving the inequality \(|s_n - L| < L/2\), two conditions emerge: \((s_n)\) could be less than \((L/2)\) above \((L)\) or less than \((L/2)\) below \((L)\), leading to the simplified conditions \((s_n > L/2)\) in both cases. Thus, we conclude that \((s_n > 0)\) for all \((n > N)\), a vital step in our example problem.
An inequality involving an absolute value, such as \(|s_n - L| < ε\), typically breaks down into two separate scenarios, capturing the 'distance' concept. It means the expression inside the absolute value can be less than \((ε)\) units above \((L)\) or less than \((ε)\) units below \((L)\).
In our case, by solving the inequality \(|s_n - L| < L/2\), two conditions emerge: \((s_n)\) could be less than \((L/2)\) above \((L)\) or less than \((L/2)\) below \((L)\), leading to the simplified conditions \((s_n > L/2)\) in both cases. Thus, we conclude that \((s_n > 0)\) for all \((n > N)\), a vital step in our example problem.
Limit of a Sequence
The limit of a sequence is a fundamental notion in calculus that embodies the value a sequence approaches as the index increases indefinitely.
When we talk about the limit, we refer to the behavior of the sequence \((s_n)\) as \((n)\) becomes very large. If \((s_n)\) approaches a specific number \((L)\) as \((n)\) goes to infinity, then \((L)\) is called the limit of the sequence. When such a limit exists, we say the sequence is convergent; otherwise, it is divergent.
In practical terms, calculating the limit usually involves determining a trend within the sequence and then formally proving that the sequence behaves as expected when taken to infinity. For the sequence in our exercise, we've utilized the definition of the limit to establish that beyond a certain point dictated by \((N)\), the terms \((s_n)\) will indeed be positive, as they close in on their limit \((L > 0)\).
When we talk about the limit, we refer to the behavior of the sequence \((s_n)\) as \((n)\) becomes very large. If \((s_n)\) approaches a specific number \((L)\) as \((n)\) goes to infinity, then \((L)\) is called the limit of the sequence. When such a limit exists, we say the sequence is convergent; otherwise, it is divergent.
In practical terms, calculating the limit usually involves determining a trend within the sequence and then formally proving that the sequence behaves as expected when taken to infinity. For the sequence in our exercise, we've utilized the definition of the limit to establish that beyond a certain point dictated by \((N)\), the terms \((s_n)\) will indeed be positive, as they close in on their limit \((L > 0)\).
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