Proving the Convergence or Divergence of the Sequence n-1/n

Understanding whether a sequence converges or diverges is a fundamental concept in calculus. The sequence in question here is ( frac{n-1}{n} ). In this article, we will explore how to determine if this sequence is convergent or divergent.

The Sequence: (frac{n-1}{n})

First, let's rewrite the given sequence in a more instructive form to understand its behavior:

(frac{n-1}{n} n - frac{1}{n})

This expression is a combination of two terms: ( n ), which grows without bound as ( n ) increases, and ( -frac{1}{n} ), which approaches zero as ( n ) increases.

Divergence Analysis

To prove whether the sequence ( frac{n-1}{n} ) diverges, we need to analyze the behavior of this sequence as ( n ) approaches infinity. We can do this by examining the limit of the sequence:

[lim_{n to infty} left(frac{n-1}{n}right) lim_{n to infty} left(n - frac{1}{n}right)]

Breaking this down into two simpler limits:

[lim_{n to infty} left(n - frac{1}{n}right) lim_{n to infty} n lim_{n to infty} left(-frac{1}{n}right)]

The term ( lim_{n to infty} n ) clearly tends to infinity as ( n ) increases:

[lim_{n to infty} n infty]

And the term ( lim_{n to infty} left(-frac{1}{n}right) ) approaches zero:

[lim_{n to infty} left(-frac{1}{n}right) 0]

Combining these two limits:

[lim_{n to infty} left(frac{n-1}{n}right) lim_{n to infty} n 0 infty]

Therefore, the sequence ( frac{n-1}{n} ) diverges to infinity as ( n to infty ).

Understanding the Behavior at Large Values of ( n )

For large values of ( n ), the term ( frac{1}{n} ) becomes very small. However, the term ( n ) becomes extremely large. To visualize this, let's consider a few examples:

Examples for Large ( n )

Suppose ( n 1000 ):

[frac{1000 - 1}{1000} frac{999}{1000} 0.999]

Suppose ( n 10000 ):

[frac{10000 - 1}{10000} frac{9999}{10000} 0.9999]

Suppose ( n 100000 ):

[frac{100000 - 1}{100000} frac{99999}{100000} 0.99999]

Even though ( frac{1}{n} ) is very small, it is overwhelmed by the large value of ( n ). This smallness of ( frac{1}{n} ) does not prevent the overall term from growing without bound as ( n ) increases.

Conclusion

Based on the analysis above, we can conclude that the sequence ( frac{n-1}{n} ) diverges to infinity as ( n to infty ).

Frequently Asked Questions

Q: Why is the sequence ( frac{n-1}{n} ) divergent?

The sequence ( frac{n-1}{n} ) is divergent because the term ( n ) grows without bound as ( n to infty ), while ( -frac{1}{n} ) approaches zero. The large value of ( n ) dominates the sequence.

Q: How can I prove the sequence ( frac{n-1}{n} ) diverges?

To prove the sequence ( frac{n-1}{n} ) diverges, we can find the limit as ( n to infty ). Since ( lim_{n to infty} left(frac{n-1}{n}right) infty ), the sequence diverges to infinity.

Q: How does the term ( frac{1}{n} ) affect the sequence?

The term ( frac{1}{n} ) approaches zero as ( n to infty ). However, its effect on ( frac{n-1}{n} ) is minimal compared to the term ( n ), which grows without bound. Therefore, the dominant term is ( n ), leading to divergence.