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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 the ratio test to find whether the following series converge or diverge: $$\sum_{n=0}^{\infty} \frac{(n !)^{3} e^{3 n}}{(3 n) !}$$

Test for convergence: $$\sum_{n=1}^{\infty} \frac{2^{n}}{n !}$$

Use the comparison test to prove the convergence of the following series: (a) \(\sum_{n=1}^{\infty} \frac{1}{2^{n}+3^{n}}\) (b) \(\sum_{n=1}^{\infty} \frac{1}{n 2^{n}}\)

Test the following series for convergence. $$\sum_{n=0}^{\infty} \frac{(-1)^{n} n}{1+n^{2}}$$

Using the methods of this section: (a) Find the first few terms of the Maclaurin series for each of the following functions. (b) Find the general term and write the series in summation form. (c) Check your results in (a) by computer. (d) Use a computer to plot the function and several approximating partial sums of the series. $$\int_{0}^{x} \cos t^{2} d t$$
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