Question

Determine the convergence of the given series. State the test used; more than one test may be appropriate. $$\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+2}\right)$$

Short answer

The series converges by the telescoping series test.

Step-by-step solution

Identify the Series Type

The given series is \( \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+2}\right) \). This is a series that appears to be a telescoping series. A telescoping series is one in which most terms cancel out with each other when expanded.

Expand the Series Terms

Expand the first few terms of the series to identify the pattern. When the series is expanded, each term is given by \( \sum \left( \frac{1}{n} - \frac{1}{n+2} \right) \). Therefore, the first few terms are: \( \left(\frac{1}{1} - \frac{1}{3}\right) + \left(\frac{1}{2} - \frac{1}{4}\right) + \left(\frac{1}{3} - \frac{1}{5}\right) + \ldots \). Notice that each positive term \( \frac{1}{n} \) will cancel with a negative term two steps later \( \frac{1}{n+2} \).

Recognize the Telescoping Pattern

In telescoping series, many terms cancel out when the individual terms are added together. In this series, the terms cancel as follows: for each \( n \), the term \( \frac{1}{n+2} \) will be subtracted when \( n+2 \) becomes the new \( n \). This leaves us with only the first few remaining terms which do not get canceled.

Determine the Remaining Terms

After cancellation, the small set of remaining terms are \( \frac{1}{1} + \frac{1}{2} \). As \( n \to \infty \), all other terms cancel out. This leaves us with the finite sum of the uncanceled terms.

Calculate the Sum of the Remaining Terms

Calculate the sum of the non-cancelled terms. In this case, it is \( \frac{1}{1} + \frac{1}{2} = 1 + 0.5 = 1.5 \).

Conclude Convergence

Since the expanded series converges to a finite value \( 1.5 \), the series \( \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+2}\right) \) converges. This conclusion is based on identifying the series as telescoping, where the next non-cancelled term appears in the sequence.

Key concepts

Series Convergence

Understanding series convergence is crucial when dealing with infinite sequences of numbers. In simple terms, a series can be thought of as the sum of the terms of a sequence. A series is said to converge if the sum of its terms gets closer and closer to a specific number as you add more terms. Conversely, if the sum grows without bound, the series diverges.
When looking at the convergence of a series, you essentially want to know if adding up all infinitely many terms gives you a finite result. Let's break it down:

• If a series converges, its partial sums will stabilize at some fixed value, say "L," as you consider more and more terms.
• When a series diverges, its partial sums could grow indefinitely or oscillate without settling down to a limit.

By examining the pattern in which terms are added, like in a telescoping series, you can determine whether a series converges or diverges.

Telescoping Pattern

The telescoping pattern is an interesting concept in calculus and series analysis. It occurs when many terms in a series cancel each other out, simplifying the evaluation of the series. This cancellation often leaves only a few terms to sum, making it much easier to determine the result.
In our example, the terms of the series are expressed as differences: \[ \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+2}\right) \]When expanded, a series like this:

• "Telescopes" by canceling out terms across the series, as each positive fraction term finds its counterpart negative in succeeding terms.
• This pattern gets its name since it "collapses" in structure, similar to how parts of a telescope slide into one another.

For our series, this cancellation results in just the initial terms remaining after simplification, highlighting the beauty of the telescoping pattern.

Infinite Series

An infinite series is a sum of infinitely many numbers that are derived from an infinite sequence. You'll often encounter a series with symbols like \( \sum_{n=1}^{\infty}a_n \).This signifies that each term comes from a sequence, and all terms are added to form a grand sum.
Infinite series can seem daunting since they involve endless addition, but various mathematical techniques help manage and understand them:

• They are central to numerous applications in calculus and real-world scenarios to model data or phenomena.
• By recognizing patterns such as in the telescoping series, mathematicians can determine if these sums stabilize at a certain value (converge) or not (diverge).

Understanding how infinite series operate is essential for solving complex mathematical problems.

Convergence Tests

Convergence tests are important tools we use to determine whether a series converges or diverges. Each test applies different criteria to assess the behavior of the series and its sums. Some common tests include the Ratio Test, Root Test, and Comparison Test, among others. These tests provide a structured way to handle the complexity of infinite series.
When dealing with a telescoping series like the one in our example, direct observation of term cancellation offers a simple "test":

• By expanding terms, observing which terms cancel, you directly reach a conclusion on the finite sum that determines convergence.
• Other tests may be more beneficial when dealing with different types of series where cancellation isn't evident.

Using convergence tests judiciously helps in predicting a series' behavior efficiently, aligning with mathematical rigor and intuition.