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Limit Definition:
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A sequence \( a_n \) converges to a limit L if for every positive number \( \epsilon \), there exists a natural number N such that for all \( n > N \), \( |a_n - L| < \epsilon \). This means the terms of the sequence get arbitrarily close to L as n increases.
The calculator evaluates the convergence using the formal definition:
Where:
Explanation: The calculator analyzes whether the terms of the sequence approach the specified limit L as n approaches infinity.
Details: Determining sequence convergence is fundamental in calculus and mathematical analysis. It helps establish the behavior of infinite series, functions, and various mathematical models in physics and engineering applications.
Tips: Enter the sequence expression (e.g., "1/n", "n/(n+1)") and the proposed limit value. The calculator will analyze whether the sequence converges to the specified limit.
Q1: What does it mean for a sequence to converge?
A: A sequence converges if its terms approach a specific finite value as the index increases without bound.
Q2: What's the difference between convergence and divergence?
A: Convergence means the sequence approaches a finite limit, while divergence means it does not approach any finite limit.
Q3: Can a sequence converge to more than one limit?
A: No, if a sequence converges, it has exactly one unique limit (the convergence is unique).
Q4: What are some common convergent sequences?
A: Examples include: \( \frac{1}{n} \to 0 \), \( \frac{n}{n+1} \to 1 \), and \( \left(1 + \frac{1}{n}\right)^n \to e \).
Q5: How is sequence convergence related to series convergence?
A: For an infinite series to converge, the sequence of its partial sums must converge. However, a sequence can converge without being the partial sums of a convergent series.