Problem 14

Question

Suppose $x_{1}:=c$ and $x_{n+1}:=x_{n}^{2}+x_{n} .$ Show that $\left\\{x_{n}\right\\}$ converges if and only if $-1 \leq c \leq 0,$ in which case it converges to 0 .

Step-by-Step Solution

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Answer

The sequence converges to 0 if \$-1 \leq c \leq 0\$, diverges otherwise.

1Step 1: Understand the Sequence

The sequence $x_n$ is defined recursively with $x_{1} = c$ and future terms given by $x_{n+1} = x_{n}^{2} + x_{n}$. We're tasked with understanding the behavior of this sequence, particularly its convergence.

2Step 2: Explore Convergence Conditions

For a sequence $x_n$ to converge, it must approach a particular value as $n ightarrow \infty$. We suspect the sequence converges to zero when $-1 \leq c \leq 0$. We need to verify this by checking behavior for different initial values of $c$.

3Step 3: Analyze Boundary Conditions

Consider the case $c = 0$: $x_1 = 0$, leading to $x_{n+1} = 0^2 + 0 = 0$. The sequence is constantly zero, hence converges to zero. Consider $c = -1$: then, $x_1 = -1$ and $x_{n+1} = (-1)^2 + (-1) = 0$. The sequence also becomes constantly zero. Next, consider $-1 < c < 0$: compute a few terms to observe convergence to zero.

4Step 4: Analyze Divergence for Other Conditions

If $c > 0$, starting with a positive $x_1$, $x_{n+1} = x_{n}^{2} + x_{n}$ will yield a strictly increasing sequence moving away from zero, thus not converging. Similarly, if $c < -1$, the sequence $x_{n+1}$ becomes more negative and also diverges.

5Step 5: Conclusion on Convergence

The sequence $x_n$ converges to zero if and only if the starting value $c$ is between $-1$ and $0$ inclusive. This is because an initial $c$ in that range keeps the terms under control, leading them to zero.

Key Concepts

Recursive SequencesConvergence ConditionsBoundary ConditionsDivergence Analysis

Recursive Sequences

A recursive sequence is one where each term is generated from the previous one using a specific rule or formula. In this context, we have a sequence $ \{x_n\} $ defined by the rule:

$ x_1 = c $
$ x_{n+1} = x_n^2 + x_n $

This means the first term is given by the initial condition $ c $, and each subsequent term depends on the square of the previous term plus the previous term itself. Recursive sequences are widely used in mathematical and computational models since they rely on previous data points to determine future behavior.
Understanding how recursive sequences develop over iterations helps in predicting the pattern and behavior of the sequence, which can include patterns like convergence or divergence.

Convergence Conditions

Convergence of a sequence means that as you continue to move forward in the sequence (as $ n \to \infty $), the terms of the sequence approach a specific value. For our sequence defined by $ x_{n+1} = x_n^2 + x_n $, the convergence condition is critical. Here, the criterion for convergence is that the initial term $ c $ must satisfy $ -1 \leq c \leq 0 $. This ensures that with each iteration, the sequence terms approach zero.

If $ c = 0 $, then $ x_1 = 0 $ and subsequent terms remain at zero, clearly converging.
If $ c = -1 $, the sequence quickly evaluates to zero.
Any $ c $ in between $-1$ and 0 also leads to terms reducing to zero.

These conditions work because the recursive formula keeps the values within a stable range, preventing divergence.

Boundary Conditions

Boundary conditions help to verify the validity of convergence within the specified limits. For the sequence $ \{x_n\} $, these are $ c = -1 $ and $ c = 0 $.
Boundary conditions are particularly useful to ensure the behavior at extreme values matches expected outcomes:

At $ c = 0 $, $ x_1 = 0 $ results in $ x_{n+1} = 0 $, a constant sequence of zeroes.
At $ c = -1 $, $ x_1 = -1 $ yields $ x_{2} = 0 $, which shifts to a sequence of zeroes thereafter.

Thus, both boundary conditions ensure that once zero is reached, the terms remain stable, indicating convergence to zero. Ensuring that boundary points meet anticipated behavior is crucial in verifying convergence claims.

Divergence Analysis

Divergence occurs when a sequence does not approach a single finite value. Instead, its terms may increase or decrease without bound. For our sequence, it diverges under certain initial conditions for $ c $:

If $ c > 0 $: Starting with a positive $ c $, the terms $ x_{n+1} = x_n^2 + x_n $ tend to grow exponentially or without bound, thus diverging away from zero.
If $ c < -1 $: The terms will move into more negative values, increasing in magnitude negatively, which means diverging similarly outside the expected range for convergence.

In both cases, the sequence fails to stabilize and thus does not converge to zero. Understanding these divergence conditions is essential to predict when a sequence becomes unbounded, depending on the initial term.

Problem 13

Problem 14

Other exercises in this chapter

Problem 13

Let s_ be the kth partial sum of $\sum x_{n}$. a) Suppose that there exists an $m \in \mathbb{N}$ such that $\lim _{k \rightarrow \infty} s_{m k}$ exists

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Problem 13

Let $\left\\{x_{n}\right\\}$ be a convergent monotone sequence. Suppose there exists a $k \in \mathbb{N}$ such that $$\lim _{n \rightarrow \infty} x_{n}=x_{

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Problem 14

Suppose $\left\\{c_{n}\right\\}$ is any sequence. Prove that for any $r \in(0,1)$ there exists a strictly increasing sequence $\left\\{n_{k}\right\\}$ of

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Problem 14

Let $\left\\{x_{n}\right\\}$ be a sequence. a) Show that $\lim x_{n}=\infty$ if and only if $\liminf x_{n}=\infty$. b) Then show that \(\lim x_{n}=-\infty

View solution