Use the Squeeze Theorem and Corollary 1.1 to prove that the...

Use the Squeeze Theorem and Corollary 1.1 to prove that the sequence converges to 0 (given that $$.\lim _{n \rightarrow \infty} \frac{1}{n}=\lim _{n \rightarrow \infty} \frac{1}{n^{2}}=0)$$.
$$a_{n}=(-1)^{n} \frac{e^{-n}}{n}$$

Key Concepts

Squeeze Theorem

The Squeeze Theorem, also known as the Sandwich Theorem, is a fundamental limit theorem which states that if a sequence (or function) is bounded between two other sequences (or functions) that converge to the same limit, then the original sequence also converges to that limit. This technique is particularly useful when dealing with sequences that oscillate or have components that make direct limit computation difficult, as it allows one to compare it with simpler sequences whose limits are already known.

Sequence Convergence

Sequence convergence is the concept where a sequence is said to converge if its terms approach a single real number as the index increases indefinitely. This process involves demonstrating that for any arbitrarily small positive number, there exists a sufficiently large index beyond which the absolute difference between the sequence terms and the limit remains less than that small number. Establishing convergence is a central task in real analysis and is crucial for understanding continuous behavior in mathematical contexts.

Exponential Decay

Exponential decay refers to the rapid decrease of quantities in which the rate of change is proportional to the current value. In sequences like the one given, the exponential term (e^(-n)) decreases very quickly as n increases, which is a key factor in ensuring that even when multiplied by another term that might not decay as fast, the overall product still approaches zero. This property is essential in comparing the given sequence with other sequences known to converge to zero.

Absolute Value and Boundedness

Analyzing the absolute value of a sequence can simplify the study of its convergence by removing any oscillatory behavior introduced by alternating signs. When the absolute value of a sequence is shown to be bounded by sequences that converge to zero, one can apply the Squeeze Theorem. This method demonstrates convergence even in sequences with alternating signs, by ensuring that the magnitude of the sequence's terms becomes arbitrarily small.

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