Problem 35 - Thomas Calculus

Question

In Exercises \(35-40,\) find a formula for the \(n\) th partial sum of the series and use it to determine if the series converges or diverges. If a series converges, find its sum. $$ \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+1}\right) $$

Step-by-Step Solution

Answer

The series converges, and its sum is 1.

Step 1: Understanding the Series

The given series is \( \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+1}\right) \). Each term in the series is of the form \( a_n = \frac{1}{n} - \frac{1}{n+1} \). This type of series is known as a telescoping series because consecutive terms cancel each other out in a pattern.

Step 2: Recognize the Telescoping Nature

To see the telescoping nature, compute the first few terms:\[a_1 = \frac{1}{1} - \frac{1}{2}\], \[a_2 = \frac{1}{2} - \frac{1}{3}\], \[a_3 = \frac{1}{3} - \frac{1}{4}\]. Notice that in the sum \((a_1 + a_2 + a_3)\), most intermediate terms cancel out, leaving only \(\frac{1}{1} - \frac{1}{4}\).

Step 3: Find the n-th Partial Sum

The n-th partial sum \(S_n\) of the series is\[S_n = \left(\frac{1}{1} - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \cdots + \left(\frac{1}{n} - \frac{1}{n+1}\right).\] This simplifies to \[S_n = 1 - \frac{1}{n+1}.\]

Step 4: Determine Convergence of the Series

To determine if the series converges, evaluate the limit of the n-th partial sum as \(n \to \infty\):\[\lim_{n \to \infty} S_n = \lim_{n \to \infty} \left(1 - \frac{1}{n+1}\right) = 1.\] Since this limit exists and is finite, the series converges.

Step 5: Conclude the Sum of the Series

Since the series converges, and the limit of the n-th partial sum \(S_n\) as \(n\) approaches infinity is 1, the sum of the series is 1.

Key Concepts

series convergence

In mathematical series analysis, understanding when a series converges is crucial. Series convergence refers to a condition where the sum of the series approaches a finite value as the number of terms increases to infinity. For a series to converge, it must continuously add up to a specific value rather than growing indefinitely.
For instance, consider the series:

• The series is expressed as an infinite sum, \( \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+1}\right) \).
• Convergence in this context means checking if the limit of its n-th partial sum exists.

In simpler terms, you look for what value the series adds up to as you include more and more terms. If the series produces a finite sum, it is convergent, as is the case in our exercise where the series converges to 1.

partial sum formula

A partial sum formula helps us find the sum of the first \( n \) terms of a series. In a telescoping series, like the one in our exercise, consecutive terms cancel out.
This can make finding the partial sum easier:

• The given series is: \( \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+1}\right) \).
• Each term \( a_n = \frac{1}{n} - \frac{1}{n+1} \) results in cancellation between successive terms.
• The n-th partial sum, \( S_n \), simplifies greatly due to this characteristic.

After canceling the intermediate terms, the formula comes down to: \[ S_n = 1 - \frac{1}{n+1} \]. This shows how telescoping series can lead to simple partial sum formulas, making it easy to compute the behavior of the series as a whole.

mathematical series analysis

Mathematical series analysis involves investigating the behavior of series, focusing on convergence, divergence, and sum computations. The series given in the exercise is a typical example of a telescoping series where the main tool is the concept of partial sums.
Key aspects include:

• Identifying series types, like telescoping, where terms cancel each other.
• Using partial sum formulas to compute the sum of a finite number of terms, simplifying more intricate calculations.
• Analyzing the limit of partial sums will determine convergence.

For the exercise, by analyzing the n-th partial sum, \( S_n = 1 - \frac{1}{n+1} \), and evaluating its limit as \( n \to \infty \), it becomes clear that the series converges.
With a final result showing the series converges to 1, series analysis provides a structured approach to solving complex mathematical problems involving infinite sums.