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Chapter 9: Problem 74

Describe the difference between \(\lim _{n \rightarrow \infty} a_{n}=5\) and \(\sum_{n=1}^{\infty} a_{n}=5\)

Short Answer

Expert verified
The statement \(\lim _{n \rightarrow \infty} a_{n}=5\) indicates that the sequence \(a_{n}\) is approaching the value of 5 as \(n\) gets extremely large, while \(\sum_{n=1}^{\infty} a_{n}=5\) indicates that when we add all terms of the sequence \(a_{n}\) starting from \(n=1\) to infinity, the sum equals 5. The former deals with the behavior of the sequence as \(n\) becomes large, and the latter deals with finding the total sum when all terms of the sequence are added together.

Step by step solution

01

Define the Limit of a Sequence

The notation \(\lim _{n \rightarrow \infty} a_{n}=5\) refers to a sequence \(a_{n}\). It means that as \(n\) approaches infinity, the values of the sequence \(a_{n}\) are getting arbitrarily close to 5.
02

Define the Sum of an Infinite Series

The notation \(\sum_{n=1}^{\infty} a_{n}=5\) refers to series. It means that if you add up all the terms of the sequence \(a_{n}\) from \(n=1\) to \(n=\infty\), the sum is 5.
03

Describe the Difference

The difference between the two notations lies in their representation. \(\lim _{n \rightarrow \infty} a_{n}=5\) refers to the limit of a sequence, i.e., the value that the terms of a sequence get arbitrarily close to as \(n\) tends to infinity, while \(\sum_{n=1}^{\infty} a_{n}=5\) refers to the sum of an infinite series, i.e., the total value you get when you add up all the terms of the sequence from \(n=1\) to \(n=\infty\).

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Key Concepts

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

Limit of a Sequence
The concept of a limit of a sequence in calculus helps us understand what happens to the terms of a sequence as the sequence progresses indefinitely. Imagine you have a sequence, denoted as \(a_n\). For this sequence, the limit describes where the terms are "heading" as the number \(n\) becomes very large. If we say \(\lim_{n \rightarrow \infty} a_n = 5\), it means that as \(n\) increases, the terms of the sequence get closer and closer to the value 5.

Here's a simple way to think about it:
  • If you were plotting these sequence values on a chart, the points would get nearer and nearer to the line \(y = 5\).
  • The limit doesn't mean the terms of the sequence are exactly 5, but rather that they are "approaching" 5.
  • This concept is crucial in determining the stability and eventual behavior of the sequence.
Infinite Series
An infinite series extends the idea of a sequence by summing infinite sequence elements. In notation, an infinite series like \(\sum_{n=1}^\infty a_n = 5\) tells us that the total of all terms in the series equals 5. This is different from the limit of a sequence since it involves summation rather than just analyzing the behavior of individual terms as in a sequence.

Understanding infinite series requires grasping:
  • Each series start from a certain point, here from \(n=1\), and ideally sums up to infinity.
  • Although the sum is an infinite process, the total can still be a finite value, such as 5 in this example.
  • Series are fundamental in representing functions or quantities that occur naturally in science and engineering.
Convergence
Convergence is a central theme when dealing with both sequences and series. In simple terms, convergence refers to the tendency of a sequence or series to approach a specific value or behavior as the terms increase indefinitely.

For a sequence, convergence means that there exists a limit. So, when we say \(\lim_{n \rightarrow \infty} a_n = 5\), we acknowledge that the sequence converges to 5.

In the realm of series, convergence involves the sum of the series approaching a finite number. As in our example, \(\sum_{n=1}^{\infty} a_n = 5\) represents a convergent series, meaning despite having an infinite number of terms, their combined sum stabilizes to 5.

  • Convergence ensures that the calculations make sense, especially when working with infinite terms.
  • It provides insight into the ultimate behavior of mathematical models.
  • A sequence or a series that does not converge is said to diverge, meaning it doesn't settle into a specific value.

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Most popular questions from this chapter

Let \(\sum a_{n}\) be a convergent series, and let \(R_{N}=a_{N+1}+a_{N+2}+\cdots\) be the remainder of the series after the first \(N\) terms. Prove that \(\lim _{N \rightarrow \infty} R_{N}=0\).

\mathrm{\\{} B e s s e l ~ F u n c t i o n ~ T h e ~ B e s s e l ~ f u n c t i o n ~ o f ~ o r d e r ~ 1 is \(J_{1}(x)=x \sum_{k=0}^{\infty} \frac{(-1)^{k} x^{2 k}}{2^{2 k+1} k !(k+1) !}\) (a) Show that the series converges for all \(x\). (b) Show that the series is a solution of the differential equation \(x^{2} J_{1}^{\prime \prime}+x J_{1}^{\prime}+\left(x^{2}-1\right) J_{1}=0 .\) (c) Use a graphing utility to graph the polynomial composed of the first four terms of \(J_{1}\). (d) Show that \(J_{0}^{\prime}(x)=-J_{1}(x)\).

Investigation The interval of convergence of the series \(\sum_{n=0}^{\infty}(3 x)^{n}\) is \(\left(-\frac{1}{3}, \frac{1}{3}\right)\) (a) Find the sum of the series when \(x=\frac{1}{6}\). Use a graphing utility to graph the first six terms of the sequence of partial sums and the horizontal line representing the sum of the series. (b) Repeat part (a) for \(x=-\frac{1}{6}\). (c) Write a short paragraph comparing the rate of convergence of the partial sums with the sum of the series in parts (a) and (b). How do the plots of the partial sums differ as they converge toward the sum of the series? (d) Given any positive real number \(M\), there exists a positive integer \(N\) such that the partial sum \(\sum_{n=0}^{N}\left(3 \cdot \frac{2}{3}\right)^{n}>M\) Use a graphing utility to complete the table. $$ \begin{array}{|l|l|l|l|l|} \hline \boldsymbol{M} & 10 & 100 & 1000 & 10,000 \\ \hline \boldsymbol{N} & & & & \\ \hline \end{array} $$

Arithmetic-Geometric Mean Iet \(a_{0}>b_{0}>0\). I.et \(a_{1}\) be the arithmetic mean of \(a_{0}\) and \(b_{0}\) and let \(b_{1}\) be the geometric mean of \(a_{0}\) and \(b_{0}\). \(a_{1}=\frac{a_{0}+b_{0}}{2} \quad\) Arithmetic mean \(b_{1}=\sqrt{a_{0} b_{0}} \quad\) Geometric mean Now define the sequences \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) as follows. \(a_{n}=\frac{a_{n-1}+b_{n-1}}{2} \quad b_{n}=\sqrt{a_{n-1} b_{n-1}}\) (a) Let \(a_{0}=10\) and \(b_{0}=3 .\) Write out the first five terms of \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\} .\) Compare the terms of \(\left\\{b_{n}\right\\} .\) Compare \(a_{n}\) and \(b_{n}\). What do you notice? (b) Use induction to show that \(a_{n}>a_{n+1}>b_{n+1}>b_{n}\), for \(a_{0}>b_{0}>0\) (c) Explain why \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) are both convergent. (d) Show that \(\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n}\).

For \(n>0\), let \(R>0\) and \(c_{n}>0 .\) Prove that if the interval of convergence of the series \(\sum_{n=0}^{\infty} c_{n}\left(x-x_{0}\right)^{n}\) is \(\left(x_{0}-R, x_{0}+R\right]\), then the series converges conditionally at \(x_{0}+R\).
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