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Chapter 4: Problem 6

\(\mathbf{F}\). Die Folgen \(\left(a_{n}\right)_{n \in \mathbb{N}}\) bzw. \(\left(b_{n}\right)_{n \in \mathbb{N}}\) seien definiert durch \(a_{n}:=\frac{(3-n)^{3}}{3 n^{3}-1}\) bzw. \(\quad b_{n}:=\frac{1+(-1)^{n} n^{2}}{2+3 n+n^{2}} \quad\) f眉r alle \(n \in \mathbb{N}\) Man entscheide bei beiden Folgen, welche der drei Eigenschaften , beschr盲nkt", 鈥瀔onvergent" bzw. , divergent" vorliegen, und man bestimme im Falle der Konvergenz den Grenzwert.

Short Answer

Expert verified
The sequence \(a_n\) is bounded and convergent with a limit of \(0\) while the sequence \(b_n\) is unbounded and divergent.

Step by step solution

01

Analyze the first sequence \(a_n\)

First, examine the sequence \(a_n:=\frac{(3-n)^{3}}{3 n^{3}-1}\). Look at the behavior of the numerator and denominator as \(n\) approaches infinity. This can be done by imagining very large numbers for \(n\) and see how the fraction behaves. This will help to determine whether the sequence is bounded, convergent or divergent.
02

Determine if \(a_n\) is bounded or not

To determine if a sequence is bounded, it must be proven if there is a number \(M\) such that every term of the sequence is less than or equal to \(M\). In this case, as \(n\) grows, the numerator and denominator both approach infinity, and the fraction approaches \(0\). Therefore, the sequence \(a_n\) is bounded.
03

Determine if \(a_n\) is convergent or divergent

Having determined that \(a_n\) is bounded, we now need to determine whether it is convergent or divergent. We can make use of the limit laws for sequences, specifically the Law of Large Numbers which states that the limit of the ratio of two sequences is the ratio of their limits, if the denominator limit is not zero. The limit of \(a_n\) as \(n\) approaches infinity is indeed \(0\). Therefore, \(a_n\) is a convergent sequence with limit \(0\).
04

Analyze the second sequence \(b_n\)

Now examine the second sequence \(b_n:=\frac{1+(-1)^{n} n^{2}}{2+3 n+n^{2}}\). As performed earlier for \(a_n\), try to determine the behavior of the numerator and denominator as \(n\) approaches infinity.
05

Determine if \(b_n\) is bounded or not

Again, we need to determine whether there exists a number \(M\) such that each term of the sequence \(b_n\) is less than or equal to \(M\). As the term (-1)^n alternates between 1 and -1, the numerator in the definition of \(b_n\) ranges from \(n^2 + 1\) to \(1 - n^2\), while the denominator grows monotonically. Hence, it isn't bounded.
06

Determine if \(b_n\) is convergent or divergent

Now determine whether the sequence \(b_n\) converges or diverges. Because the sequence is not bounded, it does not converge. However, this doesn't directly determine that the sequence diverges, it's necessary to analyze the behavior of the sequence for large values of \(n\) more closely. In this case it's clear that the sequence diverges because for even \(n\) the numerator is much larger than the denominator, while for odd \(n\) the opposite is true.
07

Conclusion of the properties of both sequences

So, the sequence \(a_n\) is bounded and convergent with limit \(0\) and the sequence \(b_n\) is unbounded and divergent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convergence and Divergence
When we talk about sequences, one key aspect to consider is whether the sequence is convergent or divergent. A sequence converges if its terms approach a specific number as the sequence progresses towards infinity. This specific number is known as the limit. On the other hand, a sequence diverges if it doesn't settle down to a particular number. Instead, it might grow indefinitely large or oscillate without heading towards any particular value.

In the exercise given, we analyzed two sequences: \(a_n = \frac{(3-n)^3}{3n^3-1}\) and \(b_n = \frac{1+(-1)^n n^2}{2+3n+n^2}\). For the sequence \(a_n\), as \(n\) becomes very large, both the numerator and denominator approach infinity, and the fraction simplifies to approach 0. Since it approaches a specific number, \(a_n\) is considered a convergent sequence with a limit of 0. Conversely, the sequence \(b_n\) has a fluctuating numerator due to \((-1)^n\) affecting whether \(n^2\) is added or subtracted, making the sequence diverge since it does not approach a single value as \(n\) increases.
Boundedness of Sequences
Understanding whether a sequence is bounded helps us in identifying its behavior in terms of convergence or divergence. A sequence is bounded if there exists a real number \(M\) such that every term of the sequence is less than or equal to \(M\). In simpler terms, the sequence doesn鈥檛 go to infinity but stays within some fixed limits.

For the sequence \(a_n\) in our exercise, it is observed that as \(n\) tends towards infinity, \(a_n\) approaches 0. This means \(a_n\) remains within the limit given by its numerator and denominator's behavior, proving it is bounded. However, the sequence \(b_n\) doesn't share this property. The term \((-1)^n n^2\) in its numerator causes significant fluctuations, potentially leading to very large positive or negative numbers as \(n\) increases. This makes \(b_n\) unbounded since there is no real number \(M\) it can consistently remain below.
Limit of Sequence
The limit of a sequence is a fundamental concept where we aim to determine what value a sequence approaches as \(n\) grows infinitely large. If a sequence does approach a specific number, that number is its limit and we can express this formally as \(\lim_{n \to \infty} a_n = L\), where \(L\) is the limit.

In examining the sequence \(a_n = \frac{(3-n)^3}{3n^3-1}\), we carefully watched how the fraction behaved as \(n\) increased. By analyzing both the numerator and the denominator, we saw that as \(n\) becomes very large, the terms simplify to yield a limit of 0. This proves that \(a_n\) converges to the limit 0. For the sequence \(b_n\), the behavior was different; \((-1)^n\ n^2\) led to an alternating or divergent behavior depending on whether \(n\) was odd or even, ultimately resulting in the failure to establish any limit. Therefore, \(b_n\) doesn鈥檛 have a limit, signifying its divergence.

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Most popular questions from this chapter

\( \mathbf{}\). Seien \(a\) und \(b\) reelle Zahlen. Die Folge \(\left(a_{n}\right)_{n \in \mathbb{N}}\) sei wie folgt rekursiv definiert: $$ a_{0}:=a, \quad a_{1}:=b, \quad a_{n}:=\frac{1}{3}\left(2 a_{n-1}+a_{n-2}\right) \quad \text { f眉r } n \geq 2 $$

\( \mathbf{}^{*}\). Es seien \(\left(a_{n}\right)_{n \in \mathbb{N}},\left(b_{n}\right)_{n \in \mathbb{N}},\left(c_{n}\right)_{n \in \mathbb{N}}\) drei reelle Zahlenfolgen mit $$ a_{n} \leq b_{n} \leq c_{n} \quad \text { fur alle } n \in \mathbb{N} $$ Man zeige: Sind \(\left(a_{n}\right)_{n \in \mathbb{N}},\left(c_{n}\right)_{n \in \mathbb{N}}\) konvergent mit $$ \lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} c_{n}=; c \in \mathbb{R} $$ so ist auch \(\left(b_{n}\right)_{n \in \mathbb{N}}\) konvergent und besitzt ebenfalls den Grenzwert \(c\).

\(\mathbf{}^{*}\), Seien \(a\) und \(b\) reelle Zahlen. Die Folge \(\left(a_{n}\right)_{n \in \mathrm{N}}\) sei wie folgt rekursiv definiert: $$ a_{0}:=a, \quad a_{1}:=b, \quad a_{n}:=\frac{1}{2}\left(a_{n-1}+a_{n-2}\right) \quad \text { f眉r } n \geq 2 $$ Man beweise, dass die Folge \(\left(a_{n}\right)_{n \in \mathbb{N}}\) konvergiert und bestimme ihren Grenzwert.

. F眉r \(x \in \mathbb{R}\) und \(n \in \mathbb{N}\) sei $$ a_{n}(x):=\left(\frac{5 x-1}{x^{2}+5}\right)^{2 n+1} $$ Man bestimme explizit die folgenden Mengen a) \(A_{1}:=\left\\{x \in \mathbb{R}:\left(a_{n}(x)\right)_{n \in \mathbb{N}}\right.\) ist nach oben beschr盲nkt \(\\}\), b) \(A_{2}:=\left\\{x \in \mathbb{R}:\left(a_{n}(x)\right)_{n \in \mathbb{N}}\right.\) ist nach unten beschr盲nkt \(\\}\), c) \(A_{3}:=\left\\{x \in \mathbb{R}:\left(a_{n}(x)\right)_{n \in \mathbb{N}}\right.\) ist nicht beschr盲nkt \(\\}\), d) \(A_{4}:=\left\\{x \in \mathbb{R}:\left(a_{n}(x)\right)_{n \in \mathbb{N}}\right.\) ist konvergent \(\\}\).

Man beweise, dass die Reihe $$ \sum_{n=1}^{\infty} \frac{1}{n(n+1)(n+2)} $$konvergiert und bestimme ihren Grenzwert,
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