Problem 51 - Thomas Calculus

Question

Find the sum of each series in Exercises \(45-52 .\) $$\sum_{n=1}^{\infty}\left(\frac{1}{\ln (n+2)}-\frac{1}{\ln (n+1)}\right)$$

Step-by-Step Solution

Answer

The sum of the series is \(-\frac{1}{\ln(2)}\).

Step 1: Identify the Type of Series

The given series can be written as \( \sum_{n=1}^{\infty} \left( a_n - a_{n+1} \right) \) where \( a_n = \frac{1}{\ln(n+1)} \). This indicates a telescoping series.

Step 2: Examine the Telescoping Nature

In a telescoping series, each term cancels with a part of the subsequent term. For the series \( \sum_{n=1}^{\infty} \left( \frac{1}{\ln(n+2)} - \frac{1}{\ln(n+1)} \right) \), substitute a few terms to observe the pattern: \( \frac{1}{\ln(3)} - \frac{1}{\ln(2)} + \frac{1}{\ln(4)} - \frac{1}{\ln(3)} + \cdots \). As noted, the series "telescopes," meaning consecutive terms cancel each other.

Step 3: Find the Sum of the Series

Because it is a telescoping series, the result of the sum depends on the uncanceled terms. The series simplifies to \(-\frac{1}{\ln(2)}\), as all other terms cancel out. So the sum of the entire series is \(-\frac{1}{\ln(2)}\).

Key Concepts

Infinite Series

An infinite series is a sum of infinitely many terms. Imagine you start with a sequence of numbers, like 1, 1/2, 1/3, and so on, and keep adding them. When you continue this without stopping, you have an infinite series.
These series can be represented in a compact form using the summation symbol: \( \sum \). This symbol tells us to add all the terms of the sequence as they go to infinity.

• Not all infinite series add up to a finite number. Some grow endlessly, while others "converge" to a specific value.
• The ability to find a sum depends on whether the series is convergent or divergent.

Caution is necessary because sometimes what looks like a large series could converge to a simple number.

Convergence

Convergence in the context of series is a fascinating concept. It's about determining whether an infinite series approaches a particular number as you add more and more terms.
When a series converges, it means the total of the series becomes closer to a specific number, no matter how many terms are added.

• In mathematics, convergence is crucial for series calculations. It helps us understand if an infinite series has a finite sum.
• A divergent series is one that doesn't have a finite sum.

If you've ever stacked blocks hoping they hold steady, that steadiness is similar to what convergence represents in math.

Natural Logarithm

The natural logarithm is a specific logarithm that uses the mathematical constant \( e \), approximately 2.718.
It is denoted as \( \ln \, x \). Using natural logarithms is common in many areas of calculus and mathematical analysis.

• The natural logarithm function is the inverse of the exponential function, meaning \( \ln(e^x) = x \).
• In finance, biology, and engineering, \( \ln \, x \) helps solve problems involving exponential growth or decay.

Natural logarithms are generally preferred due to their simplicity in calculus, especially when dealing with growth processes.

Sequence and Series

A sequence is simply a list of numbers in a specific order. Think of it as items on a row, each with a number label.
A series, on the other hand, is what you get when you add all the numbers in a sequence together.

• Sequences can be finite, with an endpoint, or infinite, going on forever.
• When you add the terms of a sequence, you get a series. This can also be finite or infinite.
• The difference between sequences and series is critical: one is a list, the other is a sum.

Understanding their interaction is essential for advanced mathematics, as seen in calculus and algebra, where they form the basis of many more complex concepts.