Problem 62 - Precalculus: Mathematical for Calculus

Question

Write the sum using sigma notation. $$\frac{1}{2 \ln 2}-\frac{1}{3 \ln 3}+\frac{1}{4 \ln 4}-\frac{1}{5 \ln 5}+\cdots+\frac{1}{100 \ln 100}$$

Step-by-Step Solution

Answer

The sum in sigma notation is \(\sum_{n=2}^{100} \frac{(-1)^{n+1}}{n \ln n}\).

Step 1: Analyze the Sequence

Examine the given sequence: \(\frac{1}{2 \ln 2}-\frac{1}{3 \ln 3}+\frac{1}{4 \ln 4}-\frac{1}{5 \ln 5}+ \cdots + \frac{1}{100 \ln 100}\). Notice that the sequence involves alternating signs with increasing denominators.

Step 2: Identify the General Term

The general term of the sequence can be written as \(a_n = \frac{(-1)^{n+1}}{n \ln n}\). This accounts for the alternating sign (due to \((-1)^{n+1}\)) and the pattern of the fractions in the sequence.

Step 3: Determine the Sigma Notation

The sequence starts at 2 and goes up to 100, so the index \(n\) starts at 2. Therefore, the sigma notation for the series is: \[ \sum_{n=2}^{100} \frac{(-1)^{n+1}}{n \ln n} \]

Step 4: Review the Sigma Expression

Ensure that the expression \(\sum_{n=2}^{100} \frac{(-1)^{n+1}}{n \ln n}\) covers all terms of the series and correctly applies to alternating signs and denominator patterns from the series.

Key Concepts

Alternating Series

An alternating series is a sequence of numbers in which the signs alternate between positive and negative. This means each term in the series switches between adding and subtracting, giving the sequence a back-and-forth pattern. In the context of our sequence

• the first term is positive,
• the second is negative,
• the third is positive,
• and so on.

This alternating nature helps determine how we write the general term of the series. We often use the expression \((-1)^{n+1}\) to ensure this alternation happens for every successive term. This pattern is crucial for capturing the essence of the sequence in mathematical notation.When creating a sigma notation of an alternating series, it’s essential to capture this sign-switching behavior right from the onset. Understanding this property is key to correctly setting up and simplifying complex series.

General Term

The general term is the formula that allows you to find any term in a sequence without writing out all the other terms first. For our exercise, the general term is represented as: \( a_n = \frac{(-1)^{n+1}}{n \ln n}\). This expression captures both the alternating signs and the specific pattern in the numerators and denominators of each term of the series.The top part, or the numerator, involves \((-1)^{n+1}\), which controls the sign. For even \(n\), the term becomes negative, and for odd \(n\), it becomes positive.The bottom part, the denominator, involves two components: the term number \(n\) and the natural logarithm of \(n\) (\(\ln n\)). This reflects the increasing complexity and size of each term as the sequence progresses.Understanding the general term is critical as it provides a concise mathematical way to express every aspect of your sequence. It’s what you look for when converting a sequence into sigma notation.

Logarithmic Function

Logarithmic functions feature prominently in mathematics due to their transformative properties, particularly in areas like growth and decay. In the exercise at hand, each term's denominator involves a natural logarithm, symbolized as \(\ln n\).The natural logarithm is the inverse operation of exponentiation for base \(e\). This unique relationship is efficient at compressing large numbers, making it highly valuable in mathematical series like the one we’re exploring.
When terms are coupled with logarithmic functions, as \(\frac{1}{n \ln n}\), they exhibit particular characteristics of growth or decay, influenced by the properties of the logarithmic function.Logarithms simplify complex calculations involving multiple terms, especially in contexts like this series where progression can be tedious to evaluate term-by-term without a basic understanding of logarithmic properties.Knowing how to manipulate and interpret logarithmic functions is invaluable in many areas of math and science, as they allow us to deal with series and calculations more effectively and intuitively.