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Chapter 10: Problem 17

Use any method to determine if the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{n^{\sqrt{2}}}{2^{n}}$$

Short Answer

Expert verified
The series converges by the Ratio Test.

Step by step solution

01

Identify the Series and Compare to a Known Series

We need to determine whether the series \( \sum_{n=1}^{\infty} \frac{n^{\sqrt{2}}}{2^n} \) converges or diverges. A good method when the terms involve exponential decay is to compare it to the geometric series. Specifically, let's consider a series of the form \( \sum_{n=1}^{\infty} r^n \). The key is to recognize that \( \frac{n^{\sqrt{2}}}{2^n} \) decays faster than any geometric series with \( r \) less than 1.
02

Apply the Ratio Test

The ratio test states that for a series \( \sum a_n \), if \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \), then the series converges. Here, \( a_n = \frac{n^{\sqrt{2}}}{2^n} \). Compute the ratio:\[ \frac{a_{n+1}}{a_n} = \frac{(n+1)^{\sqrt{2}}}{2^{n+1}} \cdot \frac{2^n}{n^{\sqrt{2}}} = \frac{(n+1)^{\sqrt{2}}}{2 \cdot n^{\sqrt{2}}} \]Simplify:\[ \frac{a_{n+1}}{a_n} = \frac{1}{2} \cdot \left(\frac{n+1}{n}\right)^{\sqrt{2}} = \frac{1}{2} \cdot \left(1 + \frac{1}{n}\right)^{\sqrt{2}} \]
03

Evaluate the Limit

Evaluate the limit as \( n \to \infty \):\[ L = \lim_{n \to \infty} \frac{1}{2} \cdot \left(1 + \frac{1}{n}\right)^{\sqrt{2}} = \frac{1}{2} \cdot 1^{\sqrt{2}} = \frac{1}{2} \]Since \( L < 1 \), by the Ratio Test, the series converges.

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Key Concepts

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

Understanding the Ratio Test
The ratio test is a powerful tool for determining the convergence of a series. It works by evaluating the limit of the ratio of successive terms. Here's how it functions:
  • Consider any series \( \sum a_n \).
  • Calculate \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
  • If \( L < 1 \), the series converges. This means the terms decrease sufficiently fast.
  • If \( L > 1 \) or the limit doesn't exist, the series diverges.
  • If \( L = 1 \), the test is inconclusive.
The beauty of this test lies in its simplicity and applicability to numerous series, helping us conclude whether a series will finish with a finite sum.
Exploring Geometric Series
A geometric series is a simple yet vital type of series, formed by multiplying each term by a fixed ratio from the previous term. It appears as:\[ \sum_{n=0}^{\infty} ar^n = a + ar + ar^2 + ar^3 + \dots \]Here, \( a \) is the first term, and \( r \) is the common ratio.
  • If \( |r| < 1 \), the series converges to \( \frac{a}{1-r} \).
  • If \( |r| \geq 1 \), the series doesn鈥檛 converge; it grows indefinitely.
Geometric series are pivotal in learning about series because they offer a clear example of when a series adds up to a distinct sum, providing a foundational concept for understanding more complex series.
What is Series Convergence?
Series convergence concerns whether a series approaches a finite sum. Different series require varied techniques to determine this. An infinite series \( \sum a_n \) can converge or diverge depending on its terms:
  • A series converges if the sum of its terms approaches a specific number as more terms are added.
  • Convergence means the series remains bounded and doesn't grow infinitely.
  • Tests like the ratio test, integral test, or direct comparison test help determine convergence.
Understanding convergence is crucial because it helps interpret whether adding infinitely many terms results in an infinite sum or a meaningful, finite value.
The Concept of Exponential Decay
Exponential decay describes a quantity decreasing at a rate proportional to its current value. This concept often applies to terms in a series:
  • Each term decreases faster as the series progresses, much like how radioactive substances halve over time.
  • In mathematical terms, if a series term looks like \( b^n \), with \( 0 < b < 1 \), it experiences exponential decay.
Exponential decay plays a significant role in certain series, primarily because such series often converge due to the rapidly decreasing terms. This rapid decrease, quicker than a steady arithmetic decline, can turn an infinite series into a calculable, finite sum.

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Most popular questions from this chapter

Prove that limits of sequences are unique. That is, show that if \(L_{1}\) and \(L_{2}\) are numbers such that \(a_{n} \rightarrow L_{1}\) and \(a_{n} \rightarrow L_{2},\) then \(L_{1}=L_{2}\).

Use a CAS to perform the following steps for the sequences. a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit \(L\) ? b. If the sequence converges, find an integer \(N\) such that \(\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) \(a_{n}=\frac{n^{41}}{19^{n}}\)

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

a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and intervals. Step 1: Plot the function over the specified interval. Step 2: Find the Taylor polynomials \(P_{1}(x), P_{2}(x),\) and \(P_{3}(x)\) at \(\bar{x}=0\) 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. \(a_{n}=\left((-1)^{n}+1\right)\left(\frac{n+1}{n}\right)\)
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