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Chapter 11: Problem 55

Prove that \(a_{n} \rightarrow 0\) iff \(\left|a_{n}\right| \rightarrow 0\)

Short Answer

Expert verified
To prove \(a_n \rightarrow 0\) iff \(|a_n| \rightarrow 0\), we first show that if \(a_n \rightarrow 0\), then \(|a_n| \rightarrow 0\) by noting that \(||a_n| - 0| = |a_n| < \epsilon\) for all \(n > N_1\). Next, we show that if \(|a_n| \rightarrow 0\), then \(a_n \rightarrow 0\) by observing \(|a_n - 0| = |a_n| < \epsilon\) for all \(n > N_2\). Thus, we conclude that \(a_n\) converges to 0 if and only if \(|a_n|\) converges to 0.

Step by step solution

01

Prove that if \(a_n \rightarrow 0\), then \(|a_n| \rightarrow 0\) ("if" part)

Assume that the sequence \(a_n\) converges to 0, which means, for any given \(\epsilon > 0\), there exists a positive integer \(N_1\) such that for all \(n > N_1\), \(|a_n - 0| < \epsilon\). In other words, \(|a_n| < \epsilon\) for all \(n > N_1\). Now, we want to show that \(|a_n|\) also converges to 0. Since we know that \(|a_n| < \epsilon\), and we also know that \(||a_n|-0|=|a_n|\), then for any \(\epsilon > 0\), we have: \[| |a_n|-0 | < \epsilon\] Therefore, \(|a_n|\) converges to 0 when \(a_n\) converges to 0.
02

Prove that if \(|a_n| \rightarrow 0\), then \(a_n \rightarrow 0\) ("only if" part)

Now let's assume that the absolute value of the elements in the sequence \(|a_n|\) converges to 0. It means, for any given \(\epsilon > 0\), there exists a positive integer \(N_2\) such that for all \(n > N_2\), \(||a_n|-0| < \epsilon\). Now, we want to show that \(a_n\) also converges to 0. Notice that \(|a_n-0| = |a_n|\). Since we know that \(||a_n| - 0| < \epsilon\), and we also know that \(|a_n| = |a_n-0|\), then for any \(\epsilon > 0\), we have: \[|a_n-0| < \epsilon\] Therefore, \(a_n\) converges to 0 when \(|a_n|\) converges to 0. Since we've shown that if \(a_n \rightarrow 0\), then \(|a_n| \rightarrow 0\) (Step 1), and if \(|a_n| \rightarrow 0\), then \(a_n \rightarrow 0\) (Step 2), we can conclude that: \[a_n \rightarrow 0 \text{ if and only if } |a_n| \rightarrow 0\]

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Key Concepts

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

Absolute Value
The concept of absolute value is foundational to understanding sequence convergence. In mathematical terms, the absolute value of a real number is the numeric value without regard to its sign. For instance, the absolute value of both 3 and -3 is 3, denoted as |3| = |-3| = 3.

When dealing with sequences, the absolute value plays a crucial role in assessing the distance of sequence terms from a point. This is because absolute value provides a measure of magnitude that's always non-negative, which is essential when discussing the limits and convergence of sequences.
Limit of a Sequence
The limit of a sequence is a fundamental concept in calculus. It is a value that the terms of the sequence get arbitrarily close to as the sequence progresses. Formally, we say that a sequence (a_n) converges to a limit L if, for any positive number 蔚 (no matter how small), there is a point in the sequence after which all terms are within 蔚 of L.

This idea is the basis for understanding how sequences behave in the long run, and whether they settle down to a single value or not. When a sequence has a limit, we use the notation a_n 鈫� L as n 鈫� 鈭�, indicating that as n becomes very large, a_n gets very close to L.
Epsilon-Delta Definition of a Limit
The 蔚-未 (epsilon-delta) definition of a limit is a precise mathematical way to define what it means for a sequence to converge to a limit. It states that a sequence (a_n) converges to a limit L if, for every 蔚 > 0, there exists a natural number N such that for all n > N, the absolute value of the difference between a_n and L is less than 蔚. Mathematically, we express this as:
  • For every 蔚 > 0
  • There exists an N such that:
  • For all n > N, |a_n - L| < 蔚
This precision allows mathematicians to communicate clearly about limits and to perform rigorous proofs regarding the behavior of sequences.
Convergence of Sequences
Convergence of sequences is the process by which the terms of a sequence approach a fixed value as the sequence progresses to infinity. A sequence is said to converge if it has a limit. In the predicate of the initial exercise, a_n 鈫� 0 and |a_n| 鈫� 0 can be understood as two sequences approaching the same limit, zero, but they approach it in terms of their magnitude and regardless of their sign because of the absolute value function.

The proof showed a reciprocal relationship; if a sequence of numbers converges to zero, so does the sequence of their absolute values, and vice versa. This is essential because it implies that convergence can be studied through the lens of absolute values, offering a simplified approach to understanding the behavior of sequences.

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Most popular questions from this chapter

(a) Show that $$\int_{-\infty}^{\infty} \frac{2 x}{1+x^{2}} d x$$ diverges by showing that $$\int_{0}^{\infty} \frac{2 x}{1+x^{2}} d x$$ diverges. (b) Then show that \(\lim _{b \rightarrow \infty} \int_{-b}^{b} \frac{2 x}{1+x^{2}} d x=0\).

A ball is dropped from a height of 100 feet. Each time it hits the ground, it rebounds to \(75 \%\) of its previous height. (a) Let \(S_{n}\) be the distance that the ball travels between the \(n\) th and the \((n+1)\) st bounce. Find a formula for \(S_{n}\). (b) Let \(T_{n}\) be the time that the ball is in the air between the \(n\) th and the \((n+1)\) st bounce. Find a formula for \(T_{n}\).

(Useful later) Let \(f\) be a continuous, positive, decreasing function defined on \((1, \infty) .\) Show that $$\int_{1}^{\infty} f(x) d x$$ converges if the sequence $$a_{n}=\int_{1}^{n} f(x) d x$$ converges.

Below we list some improper integrals. Determine whether the integral converges and, if so, evaluate the integral. $$\int_{0}^{\infty} e^{-x} \sin x d x$$

Sketch the curve, specifying all vertical and horizontal asymptotes. $$y=\sqrt{\frac{x}{x-1}}$$
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