determine whether the sequence converges and if it converges find its limit a subscript n=((-1)^n)

The sequence given is \( a_n = (-1)^n \).

Let's analyze whether the sequence converges:

### 1. Behavior of the sequence:
The term \( (-1)^n \) alternates between \( 1 \) and \( -1 \) for successive values of \( n \). Specifically:
- If \( n \) is even, \( (-1)^n = 1 \).
- If \( n \) is odd, \( (-1)^n = -1 \).

Thus, the sequence alternates as follows:
- For \( n = 1 \), \( a_1 = -1 \)
- For \( n = 2 \), \( a_2 = 1 \)
- For \( n = 3 \), \( a_3 = -1 \)
- For \( n = 4 \), \( a_4 = 1 \), and so on.

### 2. Convergence:
A sequence converges if, as \( n \) approaches infinity, the terms of the sequence approach a single fixed value (the limit).

In this case, the sequence \( a_n = (-1)^n \) does not approach a single value as \( n \to \infty \). Instead, it continues to alternate between \( 1 \) and \( -1 \), never settling on one particular value. 

Therefore, the sequence **does not converge**.

### Conclusion:
The sequence \( a_n = (-1)^n \) **does not converge**, and hence, it has no limit.

Would you like further details or have any questions?

Here are five related questions you might consider:
1. What is the definition of a convergent sequence?
2. Can a sequence that alternates between two values ever converge?
3. How do we mathematically prove that a sequence converges?
4. What is an example of a sequence that converges to a limit?
5. What is the difference between oscillating and converging sequences?

**Tip:** For a sequence to converge, the values of the sequence must get arbitrarily close to a specific number as \( n \to \infty \).

determine whether the sequence converges and if it converges find its limit a subscript n = ((-1)^n)

Explore the behavior of the sequence a_n = (-1)^n to determine whether it converges or not. Learn about oscillating sequences, convergence definitions, and why this sequence has no limit. Suitable for students in Grades 10-12.

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