Applying the Squeeze Theorem

This page demonstrates the practical application of the théorème des gendarmes (Squeeze Theorem) using the example introduced on the previous page. The sequence Un = 3 + (-1)ⁿ/n is analyzed to determine its limit as n approaches infinity.

The analysis begins by establishing inequalities:

Example: For all n ∈ N*, 3 - 1/n ≤ 3 + (-1)ⁿ/n ≤ 3 + 1/n

This inequality forms the basis for applying the Squeeze Theorem. The next step involves examining the limits of the bounding sequences:

lim(n→∞) $3 - 1/n$ = 3 lim(n→∞) $3 + 1/n$ = 3

Highlight: Both the lower and upper bounding sequences converge to 3 as n approaches infinity.

Applying the théorème des gendarmes, we can conclude that:

lim(n→∞) Un = 3

Vocabulary: Convergence - In this context, convergence refers to the property of a sequence approaching a specific value (the limit) as the index increases indefinitely.

This example effectively demonstrates how the Squeeze Theorem can be used to determine the limit of a sequence that might otherwise be challenging to evaluate directly. By "squeezing" the sequence Un between two simpler sequences that both converge to 3, we can confidently state that Un also converges to 3.

Definition: The théorème des gendarmes states that if a sequence is bounded between two sequences converging to the same limit, it also converges to that limit.

This application of the théorème des gendarmes showcases its power in solving complex limit problems by leveraging simpler, known limits.

The Squeeze Theorem (Théorème des Gendarmes)

The théorème des gendarmes, also known as the Squeeze Theorem, is a fundamental concept in mathematical analysis used to determine the limit of a sequence by comparing it with two other sequences whose limits are known. This page introduces the theorem and provides a formal definition.

Definition: Let (Un), (Vn), and (Wn) be three sequences, and I be a real number. If there exists a natural number n0 such that for all integers n ≥ n0, Vn ≤ Un ≤ Wn, and if lim(n→∞) Vn = lim(n→∞) Wn = I, then the sequence (Un) converges and lim(n→∞) Un = I.

The theorem is particularly useful when dealing with sequences that are difficult to evaluate directly. By finding two simpler sequences that "squeeze" the sequence in question, we can determine its limit.

Example: For all n ∈ N*, -1 ≤ (-1)ⁿ ≤ 1. This inequality demonstrates a simple application of the Squeeze Theorem, where (-1)ⁿ is bounded between -1 and 1.

Highlight: The théorème des gendarmes can be applied to both finite and infinite limits, making it a versatile tool in mathematical analysis.

The page concludes with the introduction of an example sequence Un defined on N* by Un = 3 + (-1)ⁿ/n, which will be analyzed using the Squeeze Theorem in the following page.