Problem 39 - Thomas Calculus

Question

In Exercises \(39-44\) , 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 to 1.

Step 1: Recognize the Series Type

The series given is \(\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+1}\right)\). This type of series is known as a telescoping series because many terms cancel out when expanded.

Step 2: Find the General Term for the Nth Partial Sum

The nth partial sum \(S_n\) is calculated by adding the first \(n\) terms of the series: \(S_n = \left(\frac{1}{1} - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \cdots + \left(\frac{1}{n} - \frac{1}{n+1}\right)\).

Step 3: Simplify the Expression for the Nth Partial Sum

In the expression for \(S_n\), observe that most intermediate terms cancel out: \[S_n = 1 - \frac{1}{n+1} \]. This is the formula for the nth partial sum.

Step 4: Determine the Limit of the Nth Partial Sum

The limit of \(S_n\) as \(n\) approaches infinity is found by taking \(\lim_{n \to \infty} (1 - \frac{1}{n+1}) = 1\).

Step 5: Conclude Convergence and Find the Sum

Since the limit of the nth partial sum exists and equals 1, the series converges and the sum of the series is \(1\).

Key Concepts

Series Convergence

When we talk about series convergence, we're trying to answer whether the sum of an infinite series results in a finite number. Convergence is crucial because it tells us if adding infinitely many terms leads to something understandable or just endlessly grows. In our example, we have a telescoping series, which is a great type of series to check for convergence. Telescoping series often have terms that cancel each other, simplifying the process.
- If the limit of the sequence of partial sums exists and is finite, the series converges. - If it does not exist or is infinite, the series diverges.
For the series defined as the limit of its partial sums is 1, meaning this series indeed converges.

Partial Sum

A partial sum is the sum of the first few terms of a series. This concept is useful because we can analyze how these sums behave as you keep adding terms. For instance, if we look at our telescoping series: the partial sum up to the nth term is given by: \[ S_n = \left(\frac{1}{1} - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \cdots + \left(\frac{1}{n} - \frac{1}{n+1}\right) \]
Most of the terms cancel out due to their structure, leaving us with a simple formula: \[ S_n = 1 - \frac{1}{n+1} \]
This simplified formula helps in determining the series' behavior as n becomes very large.

Infinite Series

An infinite series is essentially the sum of infinitely many numbers. Think of it as a way to find a single result from ongoing addition. These series are important, and often challenging to grasp, because they can reach finite limits or go on forever.
- The series here is endless because we're summing an infinite number of fractions. - Not all infinite series converge, but our telescoping series converges thanks to its structure.
Interestingly, despite having infinitely many terms, such series can often be simplified to reveal important properties.

Limit of a Sequence

Finding the limit of a sequence involves determining what value the terms of a sequence approach as the sequence progresses. When dealing with series convergence, the limit of the partial sums is crucial. In our case:

• The partial sum sequence for the given series is \( 1 - \frac{1}{n+1} \).
• As \( n \to \infty \), \( \frac{1}{n+1} \to 0 \), thus \( S_n \to 1 \).

This behavior implies that the overall series converges to the sum of 1. It's an elegant way to see how complex series can be understood by looking at simpler sequences.