To solve this, let's first explore what the notation means and how to apply it to prove that $(-1)^n$ does not converge to any real number.

Definition of Convergence

In general, a sequence $a_n$ converges to a real number $a$ if:

$\lim_{n \to \infty} a_n = a$

This means that for every positive number $\epsilon > 0$, there exists an integer $N$ such that for all $n > N$, the absolute difference between $a_n$ and $a$ is less than $\epsilon$:

$|a_n - a| < \epsilon$

In simpler terms, as $n \to \infty$, the terms of the sequence $a_n$ get arbitrarily close to the value $a$.

The Sequence $(-1)^n$

Now consider the sequence defined by $(-1)^n$, which alternates between the values $1$ and $-1$:

$(-1)^1 = -1, \quad (-1)^2 = 1, \quad (-1)^3 = -1, \quad (-1)^4 = 1, \quad \ldots$

This sequence does not "settle down" to a single value as $n \to \infty$, because it keeps switching between 1 and -1.

Showing Non-Convergence

Let's suppose, for the sake of contradiction, that $(-1)^n$ converges to some real number $a$. According to the definition of convergence, as $n \to \infty$, the terms of the sequence should get closer and closer to this value $a$.

1. Consider the subsequence where $n$ is even: $(-1)^{2k} = 1, \quad k = 1, 2, 3, \dots$ This subsequence converges to $1$.

2. Now consider the subsequence where $n$ is odd: $(-1)^{2k+1} = -1, \quad k = 1, 2, 3, \dots$ This subsequence converges to $-1$.

For the sequence $(-1)^n$ to converge to some real number $a$, both subsequences must converge to the same value. However, we see that the even subsequence converges to 1, while the odd subsequence converges to -1. This means that the sequence $(-1)^n$ does not get arbitrarily close to a single real number.

Thus, the sequence does not converge to any real number $a$.

Conclusion

The sequence $(-1)^n$ does not converge to any real number because it alternates between 1 and -1, and the limits of the subsequences (even and odd terms) do not match.

Would you like further details or have any questions?

Related Questions:

1. What is the formal definition of a convergent sequence?
2. Can you show an example of a sequence that does converge to a real number?
3. What is the difference between a divergent sequence and an oscillating sequence?
4. How does the concept of subsequences help in understanding convergence?
5. What is the importance of limits in mathematical analysis?

Tip:

When testing for convergence of a sequence, always consider the behavior of subsequences. If two subsequences of the same sequence converge to different values, the sequence itself does not converge.