Use the definition of a convergent sequence to prove the...

Use the definition of a convergent sequence to prove the following:
(a) $\frac{n-1}{2 n} \rightarrow \frac{1}{2}$ as $n \rightarrow \infty$
(b) $\frac{(-1)^{n}}{n^{2}} \rightarrow 0$ as $n \rightarrow \infty$

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Key Concepts

Convergent Sequence

A convergent sequence is one in which the terms approach a specific real number, called the limit, as the sequence progresses. In the study of sequences, demonstrating convergence involves showing that the distance between the sequence elements and the limit becomes arbitrarily small beyond a certain index.

Epsilon-N Definition

The epsilon-N definition of convergence is a formal criterion that states that for any arbitrarily small positive number ?, there exists a natural number N such that for all n greater than or equal to N, the absolute difference between the sequence term and the limit is less than ?. This rigorous approach underpins many proofs of convergence by establishing an explicit relationship between ? and the corresponding index N.

Algebraic Manipulation and Inequality Techniques

Algebraic manipulation and the use of inequalities are key techniques in proving convergence. They involve rearranging expressions, bounding terms, and establishing conditions such that the difference between the sequence terms and the limit is less than a given ?, thereby demonstrating the sequence's convergence through controlled estimates.

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