Question

A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms. $$ a_{n}=f(n)-f(n+1)-f(n+2)+f(n+3), \text { in which } f(n) \rightarrow 0 \text { as } n \rightarrow \infty . \text { Find } \sum_{n=1}^{\infty} a_{n} $$

Short answer

The sum is determined by any initial non-cancelled terms.

Step-by-step solution

Understand the Given Series

We are given a series \( a_n = f(n) - f(n+1) - f(n+2) + f(n+3) \). We need to analyze this to understand the cancelation property, which is typical of a telescoping series.

Identify Series Terms Behavior

The series is designed in a way that most terms cancel each other when summed over successive terms. As \( f(n) \rightarrow 0 \) as \( n \rightarrow \infty \), this means the function values eventually become negligible.

Expand and Simplify the Series

Let's expand the first few terms to understand the cancellation. For instance, \( a_1 = f(1) - f(2) - f(3) + f(4) \), \( a_2 = f(2) - f(3) - f(4) + f(5) \). When adding these terms, notice the middle terms start to cancel: \((f(1) - f(2) - f(3) + f(4)) + (f(2) - f(3) - f(4) + f(5)) = f(1) - f(3) - f(4) + f(5) \). Continue adding terms to notice the pattern.

Recognize the Series Collapse

Continue summing successive terms and observe that each consecutive \( f(n) \), \( f(n+1) \), and \( f(n+2) \) cancel against each other except \( f(1), -f(2), -f(3) \). As the limit goes to infinity, only non-cancelled terms remain.

Evaluate the Sum as Terms Cancel

Since terms ideally cancel due to the nature of the telescoping series, calculate the sum considering what remains after most of the terms cancel out in the infinite series. Only the initial terms left before cancellation takes over: \( f(1) \), \( -f(2) \), \( -f(3) \).

Key concepts

Infinite Series

An infinite series is a sum of an infinite sequence of terms. It can be written as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) represents each term in the series.
In this exercise, we're dealing with a specific type of series known as a telescoping series.
These kinds of series often involve terms designed to cancel out when they are added up in succession. Such cancellations simplify the infinite series, often leaving only a handful of terms that do not cancel.
Understanding the behavior of infinite series is crucial because it helps predict and evaluate the sum of a long sequence of numbers. Newton's method of approaching problems in calculus involves breaking complex problems into simpler, more manageable parts.
For infinite series, this involves understanding the pattern and behavior of the terms as \( n \) approaches infinity.
This helps in determining whether the series converges to a finite limit or diverges. In telescoping series, the sum converges due to extensive term cancellations.

Function Behavior

Function behavior refers to how a function behaves as its input values approach a particular point, usually infinity in the context of infinite series.
In the given series \( a_n = f(n) - f(n+1) - f(n+2) + f(n+3) \), the behavior of the function \( f(n) \) is important as \( n \rightarrow \infty \).
The exercise states that \( f(n) \rightarrow 0 \) as \( n \rightarrow \infty \). This indicates that the function values become negligible as \( n \) becomes very large.
Understanding the limiting behavior can assist in anticipating how the series behaves overall.
Since most series do not easily summarize to a single number, the focus becomes how each term contributes as \( n \) increases.
This is particularly useful in telescoping series, where the goal is to track what remains after extensive cancellations.

Term Cancellation

Telescoping series are defined by their property of term cancellation. This exercise illustrates how terms within a series can cancel each other out to simplify the overall sum.
Let's take a look at the structure of the series, \( a_n = f(n) - f(n+1) - f(n+2) + f(n+3) \).
When you write out successive terms like \( a_1 \) and \( a_2 \), you begin to notice a pattern of cancellation.

• For \( a_1 \), we have \( f(1) - f(2) - f(3) + f(4) \).
• For \( a_2 \), we have \( f(2) - f(3) - f(4) + f(5) \).

When these terms are summed up, many terms like \( f(2), f(3), \) and \( f(4) \) cancel out between the two terms. By continuing this process with more terms, nearly all terms cancel out, leaving only a few unchecked.
In the infinite series context, this repeated cancellation means that we're left with only the starting and effectual non-canceled bits as \( n \rightarrow \infty \).
Ultimately, understanding term cancellation can simplify solving complex series by reducing large equations to manageable forms.