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Chapter 9: Problem 53

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$a_{n}=\left(1+\frac{7}{n}\right)^{n}$$

Short Answer

Expert verified
The sequence converges to \( e^7 \).

Step by step solution

01

Recognizing the Sequence Type

The sequence given is \( a_n = \left(1 + \frac{7}{n}\right)^n \), which is a form of a known limit that approaches the exponential function as \( n \) goes to infinity.
02

Using the Exponential Limit Formula

Recall that the expression \( \left(1 + \frac{x}{n}\right)^n \) approaches \( e^x \) as \( n o \infty \). In this case, \( x = 7 \), so the sequence approaches \( e^7 \).
03

Convergence of the Sequence

Since \( \left(1 + \frac{7}{n}\right)^n \) approaches \( e^7 \) as \( n o \infty \), the sequence \( a_n \) converges.
04

Finding the Limit of the Convergent Sequence

The limit of the sequence \( a_n = \left(1 + \frac{7}{n}\right)^n \) as \( n o \infty \) is \( e^7 \). Thus, \( \lim_{n \to \infty} a_n = e^7 \).

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Key Concepts

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

Limits
Limits are a fundamental concept in calculus and analysis, providing a way to describe the behavior of sequences and functions as they approach a certain point or infinity. In the context of sequences, a limit helps us determine whether a sequence converges or diverges as its index grows very large.

  • Convergence of a sequence means that as the index \( n \) goes to infinity, the terms \( a_n \) approach a particular number, known as the limit.
  • If a sequence does not approach any specific value, it is said to diverge.
  • The notation \( \lim_{n \to \infty} a_n = L \) indicates that the sequence \( a_n \) converges to the limit \( L \).
Applying limits to sequence analysis often involves recognizing patterns or known forms that simplify the process, such as approximating certain expressions by well-known functions.
Exponential Function
The exponential function \( e^x \) is a critical part of calculus and appears frequently in sequences, integrals, and differential equations. It is defined as the limit:\[\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x\]This remarkable property allows sequences that look like \( \left(1 + \frac{c}{n}\right)^n \) to be identified immediately as those which converge to the exponent \( e^c \), given \( n \) approaches infinity.

  • In our example, \( c = 7 \) and the sequence \( \left(1 + \frac{7}{n}\right)^n \) converges to \( e^7 \).
  • This form is a classic example of how limits transform complex expressions into neat results involving the mathematical constant \( e \), which is approximately 2.71828.
  • The base of natural logarithms, \( e \), arises naturally in various growth processes, including compound interest and population models.
Understanding the role of the exponential function is key to grasping the behavior of such sequences.
Sequence Divergence
While some sequences converge to a specific value, others might not settle to a single number as \( n \) grows large, leading to divergence. A sequence diverges if it grows infinite, oscillates, or fails to approach a particular limit.

  • A common form of divergence is when terms increase or decrease without bound, signified by limits approaching positive or negative infinity.
  • Oscillating sequences, which swing back and forth without gradually settling, also represent divergence.
  • Behavior such as alternating signs or values increasing and decreasing significantly prevents the sequence from converging to a single point.
Proper assessment of a sequence's character often involves checking its general formula or employing a limit test, helping ascertain its ultimate behavior. The toolset of limits and recognition of exponential patterns are invaluable in this pursuit.

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Most popular questions from this chapter

Show that if the graph of a twice-differentiable function \(f(x)\) has an inflection point at \(x=a,\) then the linearization of \(f\) at \(x=a\) is also the quadratic approximation of \(f\) at \(x=a .\) This explains why tangent lines fit so well at inflection points.

If \(\Sigma a_{n}\) converges and \(a_{n}>0\) for all \(n,\) can anything be said about \(\Sigma\left(1 / a_{n}\right) ?\) Give reasons for your answer.

Taylor's formula with \(n=1\) and \(a=0\) gives the linearization of a function at \(x=0 .\) With \(n=2\) and \(n=3\) we obtain the standard quadratic and cubic approximations. In these exercises we explore the errors associated with these approximations. We seek answers to two questions: a. For what values of \(x\) can the function be replaced by each approximation with an error less than \(10^{-2} ?\). b. What is the maximum error we could expect if we replace the function by each approximation over the specified interval? Using a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and intervals. Step \(I:\) Plot the function over the specified interval. Step 2: Find the Taylor polynomials \(P_{1}(x), P_{2}(x),\) and \(P_{3}(x)\) at \(x=0\) Step 3: Calculate the \((n+1)\) st derivative \(f^{(n+1)}(c)\) associated with the remainder term for each Taylor polynomial. Plot the derivative as a function of \(c\) over the specified interval and estimate its maximum absolute value, \(M .\) Step 4: Calculate the remainder \(R_{n}(x)\) for each polynomial. Using the estimate \(M\) from Step 3 in place of \(f^{(n+1)}(c),\) plot \(R_{n}(x)\) over the specified interval. Then estimate the values of \(x\) that answer question (a). Step 5: Compare your estimated error with the actual error \(E_{n}(x)=\left|f(x)-P_{n}(x)\right|\) by plotting \(E_{n}(x)\) over the specified interval. This will help answer question (b). Step 6: Graph the function and its three Taylor approximations together. Discuss the graphs in relation to the information discovered in Steps 4 and 5. $$f(x)=e^{-x} \cos 2 x, \quad|x| \leq 1$$

Which of the sequences converge, and which diverge? Give reasons for your answers. $$a_{n}=\left((-1)^{n}+1\right)\left(\frac{n+1}{n}\right)$$

In each of the geometric series, write out the first few terms of the series to find \(a\) and \(r\), and find the sum of the series. Then express the inequality \(|r|<1\) in terms of \(x\) and find the values of \(x\) for which the inequality holds and the series converges. $$\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2}\left(\frac{1}{3+\sin x}\right)^{n}$$
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