Questions: Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. (If the quantity diverges, enter DIVERGES.) an = (1+(-1)^n)/n

Solution Steps

To determine the convergence or divergence of the sequence given by the \( n \)th term \( a_n = \frac{1 + (-1)^n}{n} \), we need to analyze the behavior of the sequence as \( n \) approaches infinity. The sequence alternates between two forms: \( \frac{2}{n} \) when \( n \) is even and \( \frac{0}{n} \) when \( n \) is odd. As \( n \) becomes very large, both forms approach zero. Therefore, the sequence converges to 0.

The sequence is defined as \( a_n = \frac{1 + (-1)^n}{n} \). This expression alternates based on the parity of \( n \):

• For even \( n \): \( a_n = \frac{2}{n} \)
• For odd \( n \): \( a_n = \frac{0}{n} = 0 \)

To find the limit of the sequence as \( n \) approaches infinity, we evaluate both cases:

• As \( n \to \infty \) for even \( n \): \( \frac{2}{n} \to 0 \)
• As \( n \to \infty \) for odd \( n \): \( 0 \to 0 \)

Thus, in both cases, the limit of the sequence is \( 0 \).

Since both subsequences converge to the same limit, we conclude that the sequence \( a_n \) converges to \( 0 \).

Final Answer