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A sequence of real numbers is a function from the natural numbers or a subset of natural numbers to the real numbers. Each natural number is associated with a unique real number, called the general term of the sequence.
The general term of a sequence, denoted as u_n, is a specific real number associated with each natural number n in the sequence. It defines the value of the sequence at that position.
An example of a strictly decreasing sequence is u_n = 1/n for all n in the set of natural numbers. As n increases, the value of u_n decreases.
A sequence converges to a real number ` if, for every ε > 0, there exists an N in natural numbers such that for all n ≥ N, the absolute difference |u_n - `| is less than or equal to ε.
The limit of a sequence indicates the value that the terms of the sequence approach as n becomes very large. It is denoted as lim n→+∞ u_n = `.
A bounded sequence is one where there exists a real number M such that all terms of the sequence are less than or equal to M and greater than or equal to -M.
Limit superior is the largest limit point of a sequence, while limit inferior is the smallest limit point. They help in understanding the behavior of sequences that do not converge.
Monotone sequences are sequences that are either entirely non-increasing or non-decreasing. A non-decreasing sequence satisfies u_n ≤ u_{n+1} for all n, while a non-increasing sequence satisfies u_n ≥ u_{n+1}.
A sequence is strictly increasing if for all n in natural numbers, u_{n+1} > u_n. This means each term is greater than the previous term.
A Cauchy sequence is a sequence where for every ε > 0, there exists an N such that for all m, n ≥ N, the distance |u_m - u_n| is less than ε. Cauchy sequences are important in analysis as they converge in complete spaces.
In the definition of convergence, ε represents an arbitrary small positive number that defines how close the terms of the sequence must be to the limit value as n increases.
An example of a convergent sequence is u_n = 1/n, which converges to 0 as n approaches infinity.
If a sequence converges to a limit, it is necessarily bounded. Conversely, a bounded sequence does not guarantee convergence.
A subsequence is derived from a sequence by selecting certain terms while maintaining their original order. For example, from the sequence u_n = n, the subsequence could be u_{2n} = 2n.
Natural numbers serve as the index set for sequences, allowing each term to be uniquely identified and ordered, which is essential for analyzing their properties.
When a sequence is defined on a subset of natural numbers, it means that not all natural numbers are used as indices; some may be excluded, affecting the sequence's behavior.
The first three terms of a sequence can provide insight into its behavior, such as whether it is increasing, decreasing, or oscillating, and can help in identifying patterns.
To prove that a sequence is bounded, one must show that there exists a real number M such that all terms of the sequence satisfy |u_n| ≤ M for all n in the index set.
A convergent sequence approaches a specific limit as n increases, while a divergent sequence does not approach any finite limit and may oscillate or grow indefinitely.
The notation lim n→+∞ indicates the behavior of a sequence as the index n approaches infinity, which is crucial for determining convergence and limits.
If a sequence is strictly decreasing, it implies that each term is less than the previous term, which can lead to convergence to a limit or divergence to negative infinity.