Question

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

Short answer

The series converges by the Alternating Series Test.

Step-by-step solution

Identify the series

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

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}}$$

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}.$$

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):)$

Key concepts

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.