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Chapter 7: Problem 26

Find the limit (if possible) of the sequence. \(a_{n}=5-\frac{1}{n^{2}}\)

Short Answer

Expert verified
The limit of the sequence \(a_{n}=5-\frac{1}{n^{2}}\) as \(n\) approaches to infinity is 5.

Step by step solution

01

Analyze and rewrite the sequence formula

The given sequence formula is \(a_{n}=5-\frac{1}{n^{2}}\). In order to find the limit, it is needed to subsitute \(n\) with \(\infty\).
02

Substitute \(n\) to \(\infty\) in the sequence formula

Substitute \(n\) to \(\infty\) in the sequence formula \(a_{n}\). Now, the formula becomes \(a_{n}=5-\frac{1}{\infty^{2}}\).
03

Simplify the expression

The term \(\infty^{2}\) is known to be \(\infty\). And division of any number by \(\infty\) leads to 0. Therefore simplifying the expression will lead to \(a_{n}=5-0\)
04

Find the sequence limit

Simplify the obtained expression to get the value of \(a_{n}\). Thus, \(a_{n}=5\). This is the limit of the given sequence as \(n\) approaches to infinity.

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Key Concepts

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

Understanding Infinity in Sequences
In mathematics, infinity is a concept that describes something without any bound or limit. When especially referring to sequences, such as the one given in the exercise, we tend to analyze what happens as the sequence progresses indefinitely.
Here, when the sequence formula involves a variable, such as \(n\), approaching infinity, we investigate the behavior or outcome of that sequence.
  • Infinity in a sequence not only helps identify the ultimate behavior of the sequence but also aids in determining the sequence limit.
  • With the example sequence \(a_{n}=5-\frac{1}{n^{2}}\), substituting \(n\) with infinity investigates what happens eventually, as \(n\) is infinitely increasing.
  • This substitution leads to exploring the limit which is the ultimate value the sequence approaches.
Understanding this aspect is crucial as it assists in identifying whether a sequence converges (approaches a particular value) or diverges (continues to increase or decrease indefinitely).
Simplifying Expressions to Find Limits
Simplifying expressions is a vital step in evaluating limits, especially when dealing with sequences and more complex formulas.
Simplification often involves reducing the complexity of a mathematical expression to make calculations more straightforward.
In the problem we are discussing, simplifying \(5-\frac{1}{n^{2}}\) involves recognizing the behavior of the fraction \(\frac{1}{n^{2}}\) as \(n\) increases:
  • As \(n\) approaches infinity, \(n^2\) grows tremendously, and \(\frac{1}{n^{2}}\) - a number divided by an ever-growing number - approaches zero.
  • Therefore, the expression simplifies to \(5-0\) which is just 5.
This simplification allows us to ascertain the sequence's limit effectively. It's often in these steps of algebraic manipulation and simplification that students find the clarity needed to conclude problems involving sequences.
Basics of Mathematical Sequences
Mathematical sequences are ordered lists of numbers that often follow a particular pattern or rule.
They are fundamental in various mathematical analyses because they frequently represent real-world phenomena in a simplified form.
Understanding them can sometimes require isolating the 'limit' or the value a sequence will settle into as it progresses.
  • In our exercise, the sequence \(a_{n}=5-\frac{1}{n^{2}}\) is evaluated to find where it converges as \(n\) becomes infinitely large.
  • The pattern here lies in the diminishing term \(\frac{1}{n^{2}}\), which controls how the sequence approaches a specific number, in this case, 5.
  • Thus, such sequences can be approached by initially identifying the rule (here, represented as \(a_{n}\)) and determining what happens over large \(n\).
Grasping these patterns, and understanding how they behave at their extremes, forms a key component of studies in calculus and higher-level mathematics.

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Most popular questions from this chapter

In an experiment, three people toss a fair coin one at a time until one of them tosses a head. Determine, for each person, the probability that he or she tosses the first head. Verify that the sum of the three probabilities is 1 .

The Fibonacci sequence is defined recursively by \(a_{n+2}=a_{n}+a_{n+1},\) where \(a_{1}=1\) and \(a_{2}=1\) (a) Show that \(\frac{1}{a_{n+1} a_{n+3}}=\frac{1}{a_{n+1} a_{n+2}}-\frac{1}{a_{n+2} a_{n+3}}\). (b) Show that \(\sum_{n=0}^{\infty} \frac{1}{a_{n+1} a_{n+3}}=1\).

Prove, using the definition of the limit of a sequence, that \(\lim _{n \rightarrow \infty} \frac{1}{n^{3}}=0\)

Probability In Exercises 97 and 98, the random variable \(\boldsymbol{n}\) represents the number of units of a product sold per day in a store. The probability distribution of \(n\) is given by \(P(n) .\) Find the probability that two units are sold in a given day \([P(2)]\) and show that \(P(1)+P(2)+P(3)+\cdots=1\). $$ P(n)=\frac{1}{2}\left(\frac{1}{2}\right)^{n} $$

The ball in Exercise 95 takes the following times for each fall. $$ \begin{array}{ll} s_{1}=-16 t^{2}+16, & s_{1}=0 \text { if } t=1 \\ s_{2}=-16 t^{2}+16(0.81), & s_{2}=0 \text { if } t=0.9 \\ s_{3}=-16 t^{2}+16(0.81)^{2}, & s_{3}=0 \text { if } t=(0.9)^{2} \\ s_{4}=-16 t^{2}+16(0.81)^{3}, & s_{4}=0 \text { if } t=(0.9)^{3} \end{array} $$ \(\vdots\) $$ s_{n}=-16 t^{2}+16(0.81)^{n-1}, \quad s_{n}=0 \text { if } t=(0.9)^{n-1} $$ Beginning with \(s_{2}\), the ball takes the same amount of time to bounce up as it does to fall, and so the total time elapsed before it comes to rest is given by \(t=1+2 \sum_{n=1}^{\infty}(0.9)^{n}\) Find this total time.
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