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When we talk about the "limit of a sequence," we are looking to see what value the sequence's terms get close to as they extend towards infinity. Consider the sequence given by \( a_{n} = \frac{1 + (-1)^{n}}{\sqrt{n}} \). The behavior of this sequence largely depends on whether \( n \) is even or odd.

For even values of \( n \), \((-1)^{n} = 1\), and the sequence simplifies to \( a_{n} = \frac{2}{\sqrt{n}} \). As \( n \) gets larger, \( \sqrt{n} \) increases, causing \( \frac{2}{\sqrt{n}} \) to approach 0.

• Even terms go to 0 as \( n \) grows.

On the other hand, for odd values of \( n \), \((-1)^{n} = -1\), therefore resulting in \( a_{n} = 0 \). For these terms, the limit is already at 0, which means they don't change as \( n \) increases.

• Odd terms are consistently 0.

Since both even and odd subsequences approach the same limit, the overall sequence limit is 0. In this way, we conclude: \( \lim_{{n \to \infty}} a_{n} = 0 \).

Subsequences are essentially new sequences derived by selecting certain terms from the original sequence. With \( a_{n} = \frac{1 + (-1)^{n}}{\sqrt{n}} \), we deal with two distinct subsequences based on the parity of \( n \): even and odd.

When \( n \) is even, \( a_{n} \) simplifies to \( \frac{2}{\sqrt{n}} \). This even subsequence is \( \left\{ a_2, a_4, a_6, \ldots \right\} \). As \( n \rightarrow \infty \), these even terms converge to 0 because \( \frac{2}{\sqrt{n}} \) diminishes.

Meanwhile, the odd subsequence derived when \( n \) is odd reverts to zero, i.e., \( (0, 0, 0, \ldots) \). It is purely 0 starting right from the first term and remains constant as \( n \to \infty \).

• Convergence of both even and odd subsequences to the same limit of 0 implies that the entire sequence converges.

Convergence analysis of a sequence like \( a_{n} = \frac{1 + (-1)^{n}}{\sqrt{n}} \) involves understanding how the terms behave as \( n \) increases indefinitely. For this sequence, we need to check its behavior for both even and odd indices separately.

- **Even terms**: As \( n \) becomes large, \( \frac{2}{\sqrt{n}} \) approaches 0 since the denominator grows, causing the fraction to shrink. This implies convergence to 0 for the even subsequence.

- **Odd terms**: These are constantly zero, ensuring that they are converging to 0 spontaneously.

Ultimately, the original sequence \( a_{n} \) is said to converge if every derived subsequence converges to the same value. Here, both the even and odd terms show convergence toward 0.

Because the subsequences merge into a single limit, convergence analysis confirms that the entire sequence converges to 0. By ensuring the behavior of all subsequences as \( n \to \infty \), we affirm \( \lim_{{n \to \infty}} a_{n} = 0 \).