Prove Sequence A N N1 N Is Convergent

Understanding the Convergence of Sequence A_N = N + 1/N

The study of sequences is fundamental in mathematical analysis, particularly in understanding convergence properties. A sequence ( A_N ) defined as ( A_N = N + \frac{1}{N} ) presents an interesting case for convergence. This section will explore the nature of this sequence and demonstrate whether or not it converges.

Definition of Convergence

A sequence ( A_N ) converges to a limit ( L ) if, for every positive number ( \epsilon ), there exists a natural number ( N_0 ) such that for all ( N > N_0 ), the absolute difference between ( A_N ) and ( L ) is smaller than ( \epsilon ). Mathematically, this is expressed as:

[
\forall \epsilon > 0, \exists N_0 \in \mathbb{N} : |A_N – L| < \epsilon \quad \text{for all } N > N_0
]

Evaluating the Sequence

To evaluate the convergence of the sequence ( A_N = N + \frac{1}{N} ), we consider the limit of ( A_N ) as ( N ) approaches infinity.

[
\lim_{N \to \infty} AN = \lim{N \to \infty} \left( N + \frac{1}{N} \right)
]

Breaking down the limit:

1. As ( N ) increases, ( N ) itself approaches infinity.
2. The term ( \frac{1}{N} ) approaches 0 as ( N ) becomes very large.

Therefore,

[
\lim_{N \to \infty} AN = \lim{N \to \infty} N + \lim_{N \to \infty} \frac{1}{N} = \infty + 0 = \infty
]

Conclusion Regarding Convergence

Since ( \lim_{N \to \infty} A_N = \infty ), the sequence ( A_N = N + \frac{1}{N} ) does not converge to a finite limit. Instead, it diverges to infinity.

To clarify, for any chosen limit ( L ) (where ( L ) is a finite real number), no matter how large ( N_0 ) is selected, there will always exist values of ( N ) such that ( A_N ) exceeds ( L ). Therefore, it is impossible to satisfy the convergence criteria for ( A_N ), confirming its divergence.

Understanding Divergence in Sequences

When examining the concept of divergence, it is essential to recognize that not all sequences will converge to a finite limit. A sequence can diverge in various ways: it may approach infinity, oscillate without settling at any particular value, or spiral to negative infinity. The sequence defined here exemplifies divergence to infinity, indicating rapid growth without bounds.

Application in Mathematical Analysis

The understanding of sequences and their convergence properties is significant in various fields of mathematical analysis, including calculus, real analysis, and numerical methods. Sequences are foundational in defining functions, integrals, and series. Recognizing sequences that diverge assists in identifying limits of functions and behaviors of series, which may impact convergence characteristics of summations or integrals.

FAQ

1. What is the difference between convergence and divergence in sequences?
   Convergence refers to a sequence approaching a finite limit as n approaches infinity, while divergence indicates that the sequence either grows indefinitely or does not settle at any single value.

2. Can a sequence that diverges to infinity still be bounded?
   No, a sequence that diverges to infinity is unbounded by definition, as its terms continue to grow larger without any upper limit.

3. Are there any tests to determine if a given sequence converges?
   Yes, various tests exist, such as the limit test, the ratio test, and the root test for series. For sequences, applying the limit definition of convergence is often the most straightforward method.