Lower Bound

Definition

A lower bound is a value that sets the minimum or lowest possible limit for a mathematical quantity or function. It represents the smallest value that a variable or expression can take on within a given context or set of constraints.

5 Must Know Facts For Your Next Test

1. The concept of a lower bound is crucial in the context of comparison tests, as it helps determine the convergence or divergence of a series or sequence.
2. A series or sequence is said to converge if it has a finite lower bound, and diverge if it does not have a finite lower bound.
3. The comparison test for series compares the convergence or divergence of a series to the convergence or divergence of a series with a known lower bound.
4. The comparison test for integrals compares the convergence or divergence of an integral to the convergence or divergence of an integral with a known lower bound.
5. Identifying the appropriate lower bound is essential for applying comparison tests effectively and determining the behavior of a series or integral.

Review Questions

• Explain how the concept of a lower bound is used in the context of comparison tests for series.
  • In the comparison test for series, the lower bound of a series is used to determine whether the series converges or diverges. If a series has a finite lower bound, it is said to converge, as the terms of the series are bounded below by a finite value. Conversely, if a series does not have a finite lower bound, it is said to diverge, as the terms of the series can grow without bound. The comparison test compares the convergence or divergence of a given series to the convergence or divergence of a series with a known lower bound, allowing for conclusions about the behavior of the original series.
• Describe how the concept of a lower bound is applied in the context of comparison tests for integrals.
  • The concept of a lower bound is also crucial in the context of comparison tests for integrals. When determining the convergence or divergence of an integral, the lower bound of the integral plays a key role. If an integral has a finite lower bound, it is said to converge, as the values of the integrand are bounded below by a finite value. Conversely, if an integral does not have a finite lower bound, it is said to diverge, as the values of the integrand can grow without bound. The comparison test for integrals compares the convergence or divergence of a given integral to the convergence or divergence of an integral with a known lower bound, enabling conclusions about the behavior of the original integral.
• Analyze the importance of correctly identifying the lower bound in the application of comparison tests for both series and integrals.
  • The accurate identification of the lower bound is essential for the successful application of comparison tests for both series and integrals. The lower bound determines whether a series or integral converges or diverges, which is the fundamental question addressed by comparison tests. If the lower bound is not properly recognized, the comparison test cannot be applied correctly, leading to incorrect conclusions about the behavior of the series or integral. Consequently, the ability to identify the appropriate lower bound is a critical skill in the context of comparison tests, as it directly impacts the accuracy of the analysis and the validity of the conclusions drawn about the convergence or divergence of the mathematical expressions under investigation.

Related terms

Upper Bound: An upper bound is a value that sets the maximum or highest possible limit for a mathematical quantity or function.

Infimum: The infimum, or greatest lower bound, is the largest value that is less than or equal to all elements in a set.

Supremum: The supremum, or least upper bound, is the smallest value that is greater than or equal to all elements in a set.