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Chapter 1: Problem 5

Test the following series for convergence. $$\sum_{n=2}^{\infty} \frac{(-1)^{n}}{\ln n}$$

Short Answer

Expert verified
The series converges by the Alternating Series Test.

Step by step solution

01

Identify the series

Identify the given series: the series is the alternating series $$\frac{(-1)^{n}}{\text{ln} n}$$
02

Apply the Alternating Series Test conditions

The Alternating Series Test (Leibniz test) states that a series of the form $$\text{$$\text{$$\text{$$\text{\text{'Leibniz test'}{-1}}^{n}b_{n}$$$$will converge if:$$1.$$ The terms $$\text{b}_{n}$$ decrease monotonically: \text{ \text{if \text{}{ln }): The natural logarithm function, uniformly }ormally increases. To prove the terms decrease, rewrite as: $$b_{n} = \frac{1}{\text{ln} {n}}$$
03

Check if $$1/\text{ln}$$ n decreases

Show $$1/\text{ln}$$ n is decreasing. Consider the derivative of $$ b_{n} = \frac{1}{\text{ln}{n}}: interpret as decreasing: \text{1:$$ - (\text{ln}(\text{}{n} is decreasing for} {n\text=>=2}.$$
04

Check the limit condition

Evaluate $$\text{}$$ Evaluate the limit condition:$$$\text{If both are}$$\text{}else converging:\text{=0}):$$$Step (The series fulfills): These conditions:: The limit conditions have been fulfilled):)$

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Key Concepts

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

Series Convergence
In mathematics, a series is the sum of the terms of a sequence of numbers. To determine if a series converges means figuring out if adding the sequence's terms approaches a specific value as more terms are included. Understanding series convergence is crucial because it tells us if an infinite sum can result in a finite value.
Leibniz Test
The Leibniz test, also known as the Alternating Series Test, is used to check if an alternating series converges. An alternating series alternates between positive and negative terms. To apply the Leibniz test, two main conditions need to be satisfied:
Natural Logarithm
The natural logarithm, denoted as \(\text{ln}(x)\), is the logarithm to the base \(e\) (where \(e \approx 2.71828\)). It helps us understand growth processes, such as compound interest. Understanding the properties of natural logarithms is helpful in analyzing functions like those seen in series convergence tests.

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Most popular questions from this chapter

Use Maclaurin series to do and check your results by computer. $$\lim _{x \rightarrow 0}\left(\frac{1+x}{x}-\frac{1}{\sin x}\right)$$

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations. $$\lim _{x \rightarrow 0} \frac{1-e^{x^{3}}}{x^{3}}$$

Find the interval of convergence of each of the following power series; be sure to investigate the endpoints of the interval in each case. $$\sum_{n=1}^{\infty} \frac{n}{n+1}\left(\frac{x}{3}\right)^{n}$$

The energy of an electron at speed \(v\) in special relativity theory is \(m c^{2}\left(1-v^{2} / c^{2}\right)^{-1 / 2}\) where \(m\) is the electron mass, and \(c\) is the speed of light. The factor \(m c^{2}\) is called the rest mass energy (energy when \(v=0\) ). Find two terms of the series expansion of \(\left(1-v^{2} / c^{2}\right)^{-1 / 2},\) and multiply by \(m c^{2}\) to get the energy at speed \(v\). What is the second term in the energy series? (If \(v / c\) is very small, the rest of the series can be neglected; this is true for everyday speeds.)

Test the following series for convergence or divergence. Decide for yourself which test is easiest to use, but don't forget the preliminary test. Use the facts stated above when they apply. $$\sum_{n=2}^{\infty}(-1)^{n} \frac{n}{n-1}$$
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