Problem 5 - Thomas Calculus

Question

In Exercises \(1-6,\) find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$ \frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\cdots+\frac{1}{(n+1)(n+2)}+\cdots $$

Step-by-Step Solution

Answer

The series converges and its sum is \( \frac{1}{2} \).

Step 1: Recognize the pattern

Each term in the series can be represented as \( \frac{1}{(n+1)(n+2)} \). This expression suggests a telescoping nature, which can be simplified.

Step 2: Simplify the general term

Simplify the general term \( \frac{1}{(n+1)(n+2)} \) using partial fraction decomposition. This can be written as:\[\frac{1}{(n+1)(n+2)} = \frac{1}{n+1} - \frac{1}{n+2}\].

Step 3: Write the n-th partial sum

Write the expression for the nth partial sum as the sum of each simplified term:\[S_n = \left( \frac{1}{2} - \frac{1}{3} \right) + \left( \frac{1}{3} - \frac{1}{4} \right) + \cdots + \left( \frac{1}{n+1} - \frac{1}{n+2} \right)\]

Step 4: Observe the telescoping nature

Notice that most terms in the series cancel out, leaving only the first term of the first fraction and the second term of the last fraction:\[S_n = \frac{1}{2} - \frac{1}{n+2}\]

Step 5: Determine the convergence

Examine the limit of the n-th partial sum as \( n \to \infty \):\[\lim_{{n \to \infty}} S_n = \lim_{{n \to \infty}} \left( \frac{1}{2} - \frac{1}{n+2} \right) = \frac{1}{2}\]Thus, the series converges to \( \frac{1}{2} \).

Key Concepts

Telescoping Series

A telescoping series is a special type of series where many terms cancel out when summed up. This cancellation occurs due to the nature of successive terms. Understanding telescoping series is crucial because it helps simplify complex series into manageable expressions.
In the given problem, each term of the series can be written as \( \frac{1}{(n+1)(n+2)} \). After using partial fraction decomposition, this term becomes \( \frac{1}{n+1} - \frac{1}{n+2} \). This form highlights the telescoping nature because:

• When adding up multiple terms, most intermediate terms cancel.
• Only a few terms from the beginning and the end of the series remain.

In our example, after summing terms and simplifying the expression, all but the first and last terms disappear, helping us find the sum easily.

Convergence of Series

The convergence of a series is about determining whether the series adds up to a finite number as the number of terms goes to infinity. For a series to converge, its partial sums must approach a specific limit.
In this exercise, the series is expressed as a telescoping series, allowing the partial sums to be easily calculated. The nth partial sum, \( S_n = \frac{1}{2} - \frac{1}{n+2} \), displays this pattern. As \( n \to \infty \), the term \( \frac{1}{n+2} \) approaches zero, leaving:

• \( \lim_{{n \to \infty}} S_n = \frac{1}{2} \).

This result confirms that the series converges to \( \frac{1}{2} \), making it both simple and vivid after understanding the telescoping effect.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express complex rational expressions as a sum of simpler fractions. This method is especially useful when working with series to reveal patterns such as telescoping.
To perform partial fraction decomposition, you break down a fraction like \( \frac{1}{(n+1)(n+2)} \) into simpler components. For the given fraction, we express it as:

• \( \frac{1}{(n+1)(n+2)} = \frac{1}{n+1} - \frac{1}{n+2} \).

This step simplifies understanding and calculation, as it turns the series into a form where terms can cancel out. Recognizing and applying partial fraction decomposition is a crucial skill for simplifying and solving series, making it an indispensable tool in tackling complex mathematical problems efficiently.