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Chapter 8: Problem 20

Use the Root Test to determine whether the following series converge. $$\sum_{k=1}^{\infty}\left(\frac{2 k}{k+1}\right)^{k}$$

Short Answer

Expert verified
Answer: The series diverges.

Step by step solution

01

To apply the Root Test, we will find the limit of the \(k\)th root of the absolute value of each term, as k approaches infinity: $$\lim_{k \to \infty} \left|\left(\frac{2 k}{k+1}\right)^{k}\right|^{\frac{1}{k}}$$ #Step 2: Simplify the expression#

We can simplify the expression as follows: $$\lim_{k \to \infty} \left(\left|\frac{2 k}{k+1}\right|\right)^{k\cdot\frac{1}{k}}$$ Since \(k>0\), the absolute value is not necessary. Further, we can see that \(k \cdot \frac{1}{k} = 1\). This simplifies our expression to: $$\lim_{k \to \infty} \left(\frac{2 k}{k+1}\right)$$ #Step 3: Calculate the limit#
02

To calculate the limit, we can divide each term in the fraction by the highest power of \(k\) in the denominator, which is \(k\): $$\lim_{k \to \infty} \left(\frac{2}{1+\frac{1}{k}}\right)$$ As \(k\) approaches infinity, the term \(\frac{1}{k}\) approaches zero. Thus, the limit becomes: $$\lim_{k \to \infty} \left(\frac{2}{1}\right)=2$$ #Step 4: Determine convergence or divergence using the Root Test result#

Since the limit calculated in Step 3 is greater than 1, the Root Test tells us that the given series diverges: $$\sum_{k=1}^{\infty}\left(\frac{2 k}{k+1}\right)^{k}$$ is a divergent series.

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Key Concepts

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

Series Convergence
When analyzing if a series converges or not, one of our main tools is the convergence test. Imagine a series as an infinite sum of numbers. We want to know if this sum eventually settles down to a fixed value (converges) or keeps increasing indefinitely (diverges).
The Root Test is a popular method for checking convergence. It's particularly useful for series that include terms raised to a power, like the series we are discussing.
To apply the Root Test, we take the following steps:
  • Identify the general term of the series.
  • Take the limit of the nth root of the absolute value of the term, as n approaches infinity.
  • Use the resulting value to assess the convergence.
If the result is less than 1, the series converges absolutely. If it is greater than 1, the series diverges. This approach can sometimes simplify the job of determining convergence, especially when other methods might require complicated algebra.
Divergent Series
A series is called divergent if the sum of its terms grows without bound as you add more and more of them together. The intriguing part of divergent series is that sometimes, they might seem like they are growing to a limit, but if scrutinized, they don't settle to a fixed value.
Divergent series are critical in many fields of mathematics because they help identify sequences or phenomena that do not stabilize, which can be essential for defining the behavior of more complex systems.
In the Root Test, once we evaluated and determined that the limit of our series was 2, we concluded that the series diverges. This is because, according to the Root Test criterion, any limit greater than 1 points to divergence. So, for our series, which becomes unbounded, the sum doesn't approach any fixed value, thus confirming divergence.
Limit of a Sequence
The concept of a limit is central to calculus and analysis, especially when dealing with sequences and series. In simple terms, the limit of a sequence is a value that the elements of the sequence get closer to, as the sequence progresses to infinity.
In the context of the Root Test, calculating the limit of a sequence involves examining the behavior of each term as the index increases towards infinity. For the series that was tested, we simplified it to evaluate the limit as part of applying the Root Test. We computed the limit of our series' general term by transforming it into a simpler expression.
Situations where the limit equals a particular constant (like the 2 in our case) help determine specific properties about the sequence or its summative series.
  • If the limit is 0, the sequence converges to this fixed value.
  • If the limit is a non-zero constant, this can suggest divergence based on the context, as seen in the Root Test.
  • If there is no finite limit, the sequence diverges.
Understanding the limit of a sequence is a vital step in assessing the convergence or divergence of the whole series.

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Most popular questions from this chapter

Repeated square roots Consider the expression \(\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}},\) where the process continues indefinitely. a. Show that this expression can be built in steps using the recurrence relation \(a_{0}=1, a_{n+1}=\sqrt{1+a_{n}}\), for \(n=0,1,2,3, \ldots . .\) Explain why the value of the expression can be interpreted as \(\lim _{n \rightarrow \infty} a_{n},\) provided the limit exists. b. Evaluate the first five terms of the sequence \(\left\\{a_{n}\right\\}\) c. Estimate the limit of the sequence. Compare your estimate with \((1+\sqrt{5}) / 2,\) a number known as the golden mean. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. e. Repeat the preceding analysis for the expression \(\sqrt{p+\sqrt{p+\sqrt{p+\sqrt{p+\cdots}}},}\) where \(p>0 .\) Make a table showing the approximate value of this expression for various values of \(p .\) Does the expression seem to have a limit for all positive values of \(p ?\)

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\cos \left(0.99^{n}\right)+\frac{7^{n}+9^{n}}{63^{n}}$$

A ball is thrown upward to a height of \(h_{0}\) meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let \(h_{n}\) be the height after the nth bounce and let \(S_{n}\) be the total distance the ball has traveled at the moment of the nth bounce. a. Find the first four terms of the sequence \(\left\\{S_{n}\right\\}\) b. Make a table of 20 terms of the sequence \(\left\\{S_{n}\right\\}\) and determine \(a\) plausible value for the limit of \(\left\\{S_{n}\right\\}\) $$h_{0}=20, r=0.5$$

Pick two positive numbers \(a_{0}\) and \(b_{0}\) with \(a_{0}>b_{0},\) and write out the first few terms of the two sequences \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}:\) $$a_{n+1}=\frac{a_{n}+b_{n}}{2}, \quad b_{n+1}=\sqrt{a_{n} b_{n}}, \quad \text { for } n=0,1,2 \dots$$ (Recall that the arithmetic mean \(A=(p+q) / 2\) and the geometric mean \(G=\sqrt{p q}\) of two positive numbers \(p\) and \(q\) satisfy \(A \geq G.)\) a. Show that \(a_{n} > b_{n}\) for all \(n\). b. Show that \(\left\\{a_{n}\right\\}\) is a decreasing sequence and \(\left\\{b_{n}\right\\}\) is an increasing sequence. c. Conclude that \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) converge. d. Show that \(a_{n+1}-b_{n+1}<\left(a_{n}-b_{n}\right) / 2\) and conclude that \(\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n} .\) The common value of these limits is called the arithmetic-geometric mean of \(a_{0}\) and \(b_{0},\) denoted \(\mathrm{AGM}\left(a_{0}, b_{0}\right)\). e. Estimate AGM(12,20). Estimate Gauss' constant \(1 / \mathrm{AGM}(1, \sqrt{2})\).

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty} \frac{3}{10^{k}}$$
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