Problem 5
Question
zeigen Sie, da\beta die Folge \(\left(a_{n}\right), a_{n}:=1-\frac{1}{n}\) nach oben beschränkt und monoton wachsend ist.
Step-by-Step Solution
Verified Answer
The sequence is bounded above by 1 and is monotonically increasing.
1Step 1: Understanding the Sequence
The sequence given is \(a_n = 1 - \frac{1}{n}\). Here, \(n\) is a natural number and as \(n\) increases, the term \(\frac{1}{n}\) decreases.
2Step 2: Determine the Upper Bound
Notice that for any natural number \(n\), the term \(\frac{1}{n}\) will always be a positive number. Therefore, \(1 - \frac{1}{n}\) will always be less than 1. Thus, 1 is an upper bound for this sequence.
3Step 3: Check Monotonicity
To check if the sequence is monotonically increasing, compare \(a_n\) with \(a_{n+1}\). Compute \(a_{n+1} = 1 - \frac{1}{n+1}\) and notice that \(a_{n+1} > a_n\) because \(\frac{1}{n+1} < \frac{1}{n}\), implying \(1 - \frac{1}{n+1} > 1 - \frac{1}{n}\). Therefore, the sequence is monotonically increasing.
4Step 4: Conclusion
The sequence \(a_n = 1 - \frac{1}{n}\) is bounded above by 1 and is monotonically increasing as \(n\) increases.
Key Concepts
Upper Bound in SequencesMonotonicity of SequencesNatural Numbers in Sequences
Upper Bound in Sequences
In mathematics, an upper bound of a sequence is a value that the sequence will not exceed. For the sequence \(a_n = 1 - \frac{1}{n}\), understanding its upper bound is crucial to analyzing its behavior. Here, the sequence includes expressions involving the natural numbers as denominators, meaning \(n = 1, 2, 3, \ldots\).
When we calculate \(a_n\) for these numbers, \(\frac{1}{n}\) becomes smaller as \(n\) becomes larger. Consequently, the expression \(1 - \frac{1}{n}\) approaches but never surpasses 1. This characteristic makes 1 the upper bound. All terms in the sequence will always remain less than or equal to 1, confirmed by the fact that \(1 - \frac{1}{n} < 1\) for all natural numbers \(n\).
Understanding an upper bound helps us grasp the limits to a sequence's growth. It sets the maximum ceiling a sequence is allowed to climb to, ensuring predictability and control in various mathematical analyses.
When we calculate \(a_n\) for these numbers, \(\frac{1}{n}\) becomes smaller as \(n\) becomes larger. Consequently, the expression \(1 - \frac{1}{n}\) approaches but never surpasses 1. This characteristic makes 1 the upper bound. All terms in the sequence will always remain less than or equal to 1, confirmed by the fact that \(1 - \frac{1}{n} < 1\) for all natural numbers \(n\).
Understanding an upper bound helps us grasp the limits to a sequence's growth. It sets the maximum ceiling a sequence is allowed to climb to, ensuring predictability and control in various mathematical analyses.
Monotonicity of Sequences
Monotonicity describes how a sequence consistently increases or decreases. For sequence \(a_n = 1 - \frac{1}{n}\), its behavior is determined by comparing successive terms, such as \(a_n\) and \(a_{n+1}\).
To see if it is monotonically increasing or decreasing, one must evaluate \(a_{n+1} - a_n\). In this case, \(a_{n+1} = 1 - \frac{1}{n+1}\) and \(a_n = 1 - \frac{1}{n}\). Because \(\frac{1}{n+1} < \frac{1}{n}\), it follows that \(1 - \frac{1}{n+1} > 1 - \frac{1}{n}\), indicating \(a_{n+1} > a_n\).
Therefore, the sequence is said to be monotonically increasing since each term is larger than the one before. Monotonicity is a key feature influencing how sequences converge. Whether increasing or decreasing, knowing monotonicity helps predict the eventual behavior of sequences in larger mathematical contexts.
To see if it is monotonically increasing or decreasing, one must evaluate \(a_{n+1} - a_n\). In this case, \(a_{n+1} = 1 - \frac{1}{n+1}\) and \(a_n = 1 - \frac{1}{n}\). Because \(\frac{1}{n+1} < \frac{1}{n}\), it follows that \(1 - \frac{1}{n+1} > 1 - \frac{1}{n}\), indicating \(a_{n+1} > a_n\).
Therefore, the sequence is said to be monotonically increasing since each term is larger than the one before. Monotonicity is a key feature influencing how sequences converge. Whether increasing or decreasing, knowing monotonicity helps predict the eventual behavior of sequences in larger mathematical contexts.
Natural Numbers in Sequences
The concept of natural numbers is foundational in defining sequences. These are the numbers \(1, 2, 3, \dots\), and are used extensively in mathematical sequences and series. In the sequence \(a_n = 1 - \frac{1}{n}\), \(n\) specifically is a natural number, serving as the index.
Using natural numbers as indices ensures sequences are defined and calculated in a straightforward manner. Each increasing natural number \(n\) shapes how the terms of the sequence evolve.
Using natural numbers as indices ensures sequences are defined and calculated in a straightforward manner. Each increasing natural number \(n\) shapes how the terms of the sequence evolve.
- They provide a clear and logical ordering – crucial for understanding progression.
- These numbers keep sequences intuitive for mathematicians to analyze, as they present a chronological system to follow.
Other exercises in this chapter
Problem 3
Geben Sie ein Beispiel an für eine Folge, die nach oben, aber nicht nach unten beschränkt ist.
View solution Problem 4
Prüfen Sie nach, ob die Folge \(\left(a_{n}\right), a_{n}=2^{n}\) aus Aufgabe 1 monoton wachsend oder monoton fallend ist.
View solution Problem 10
Bestimmen Sie die Grenzwerte (1) \(\lim _{n \rightarrow \infty} \frac{6 n^{2}+5 n}{4 n^{2}+n+1}\), (2) \(\frac{1 i m}{n \rightarrow \infty} \frac{8 n^{5}+9 n^{3
View solution