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Chapter 3: Problem 16

Prove rigorously that \(\sqrt{n+1}-\sqrt{n} \rightarrow 0\) as \(n \rightarrow+\infty\)

Short Answer

Expert verified
\(\sqrt{n+1} - \sqrt{n} \to 0\) as \(n \to \infty\) because the denominator grows infinitely large.

Step by step solution

01

Define the Difference

Start by defining the expression we want to analyze: \[ \sqrt{n+1} - \sqrt{n} \]. We aim to show that this difference approaches 0 as \( n \) approaches infinity.
02

Rationalize the Expression

Multiply and divide by the conjugate to rationalize the expression:\[ \sqrt{n+1} - \sqrt{n} = \frac{(\sqrt{n+1} - \sqrt{n})(\sqrt{n+1} + \sqrt{n})}{\sqrt{n+1} + \sqrt{n}} = \frac{n+1 - n}{\sqrt{n+1} + \sqrt{n}} \]. This simplifies to \[ \frac{1}{\sqrt{n+1} + \sqrt{n}} \].
03

Analyze the Denominator

As \( n \to \infty \), observe that both \( \sqrt{n+1} \) and \( \sqrt{n} \) approach infinity. Therefore:\[ \sqrt{n+1} + \sqrt{n} \] is a large number.
04

Estimate the Limit

Since the denominator \( \sqrt{n+1} + \sqrt{n} \) grows without bound as \( n \to \infty \), the expression:\[ \frac{1}{\sqrt{n+1} + \sqrt{n}} \] tends to 0. Thus, the original difference \( \sqrt{n+1} - \sqrt{n} \) approaches 0 as \( n \to \infty \).
05

Conclusion

We have shown that the expression \[ \frac{1}{\sqrt{n+1} + \sqrt{n}} \] tends to 0. Hence:\[ \sqrt{n+1} - \sqrt{n} \to 0 \]as \( n \to \infty \). This completes the proof that \( \sqrt{n+1} - \sqrt{n} \rightarrow 0 \) as \( n \rightarrow \infty \).

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Key Concepts

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

Rationalization
Rationalization is a mathematical technique often used to simplify expressions that contain roots or radicals. This process involves multiplying the numerator and the denominator by a conjugate or a suitable expression that helps in removing the radicals from the denominator.
  • The conjugate of an expression like \( \sqrt{a} - \sqrt{b} \) is \( \sqrt{a} + \sqrt{b} \).
  • By multiplying these expressions, the radicals in the denominator typically cancel out, leaving a more manageable expression.
In the expression \( \sqrt{n+1} - \sqrt{n} \), rationalization helps in expressing it as \( \frac{1}{\sqrt{n+1} + \sqrt{n}} \) by multiplying both the numerator and denominator by the conjugate, \( \sqrt{n+1} + \sqrt{n} \). This step is crucial as it turns the problem into a form that is easier to analyze when discussing limits.
Infinity
Infinity is a concept in mathematics that describes something without any bound or end. It is often used to describe the behavior of sequences or functions as inputs become very large.
  • When we say \( n \to \infty \), we mean that \( n \) is increasing without any fixed end point.
  • As \( n \) increases, expressions containing \( n \) such as \( \sqrt{n+1} \) and \( \sqrt{n} \) approach infinity.
In the context of our problem, examining the behavior of \( \sqrt{n+1} + \sqrt{n} \) as \( n \to \infty \) helps us understand how the entire expression simplifies. As both terms become very large, they form a large denominator, influencing the value of our rationalized expression. Essentially, understanding infinity helps predict the behavior of components as they grow without bound.
Limit Estimation
Limit estimation is a technique used to determine what value a function approaches as the input grows larger or smaller. It is especially helpful in sequences and series to find their eventual behavior.
  • In this exercise, we aim to estimate the limit of \( \sqrt{n+1} - \sqrt{n} \) as \( n \to \infty \).
  • By rewriting the expression as \( \frac{1}{\sqrt{n+1} + \sqrt{n}} \), we are able to estimate how this fraction behaves.
As \( \sqrt{n+1} + \sqrt{n} \) becomes infinitely large, \( \frac{1}{\sqrt{n+1} + \sqrt{n}} \) tends towards zero. Therefore, the original difference \( \sqrt{n+1} - \sqrt{n} \) also approaches zero. Understanding limit estimation helps in concluding that certain mathematical expressions converge (or reach) a particular value, such as zero in this problem.

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Most popular questions from this chapter

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