Questions: To which of the following does the sequence xn, where xn=((-1)^n+n)/n, converge? 0 -1 1 Does not converge

Transcript text: To which of the following does the sequence $\left\{x_{n}\right\}$, where $x_{n}=\frac{(-1)^{n}+n}{n}$, converge? 0 $-1$ 1 Does not converge

Solution

Solution Steps

Step 1: Analyze the Sequence

The sequence given is $ x_n = \frac{(-1)^n + n}{n} $. We need to determine the limit of this sequence as $ n \to \infty $.

Step 2: Simplify the Expression

Let's simplify the expression for $ x_n $:

\[ x_n = \frac{(-1)^n + n}{n} = \frac{(-1)^n}{n} + \frac{n}{n} = \frac{(-1)^n}{n} + 1 \]

Step 3: Evaluate the Limit

We need to evaluate the limit:

\[ \lim_{n \to \infty} x_n = \lim_{n \to \infty} \left( \frac{(-1)^n}{n} + 1 \right) \]

The term $\frac{(-1)^n}{n}$ oscillates between $\frac{1}{n}$ and $-\frac{1}{n}$. As $ n \to \infty $, both $\frac{1}{n}$ and $-\frac{1}{n}$ approach 0. Therefore, the limit of $\frac{(-1)^n}{n}$ is 0.

Thus, the limit of the sequence is:

\[ \lim_{n \to \infty} x_n = 0 + 1 = 1 \]