Problem 19 - Discrete Mathematics with Applications

Question

Sums of the form \(S=\sum_{i=m+1}^{n}\left(a_{i}-a_{i-1}\right)\) are telescoping sums. Show that \(S=a_{n}-a_{m} .\)

Step-by-Step Solution

Answer

In summary, the telescoping sum can be demonstrated as follows: 1. Write down the given telescoping sum \(S=\sum_{i=m+1}^{n}\left(a_{i}-a_{i-1}\right)\). 2. Expand the sum by writing down the first few terms, which results in various cancellations. 3. Observe that most of the terms cancel each other out, and simplify the sum. Hence, we've shown that the sum \(S=\sum_{i=m+1}^{n}\left(a_{i}-a_{i-1}\right) = a_{n}-a_{m}\).

Step 1: Write down the sum

Write down the given telescoping sum: \[S=\sum_{i=m+1}^{n}\left(a_{i}-a_{i-1}\right)\].

Step 2: Expand the sum

Expand the sum by writing down the first few terms: \[S = (a_{(m +1)} - a_{m}) + (a_{(m+2)} - a_{(m+1)}) + (a_{(m+3)} - a_{(m+2)}) + \cdots + (a_{n} - a_{(n-1)}).\]

Step 3: Simplify the sum

Observe that most of the terms cancel each other out. In the expanded sum, you can see that: - \(a_{(m + 1)}\) is canceled by \(-a_{(m+1)}\) in the second term, - \(a_{(m+2)}\) is canceled by \(-a_{(m+2)}\) in the third term, - \(a_{(m+3)}\) is canceled by \(-a_{(m+3)}\) in the fourth term, and so on. This pattern continues until the second last term, \(a_{(n - 1)}\), is canceled by \(-a_{(n-1)}\) in the last term. Now we can write a simplified expression for the sum: \[S = a_{n} - a_{m}.\]

Step 4: Conclusion

In conclusion, we have shown that the sum \(S=\sum_{i=m+1}^{n}\left(a_{i}-a_{i-1}\right)\) is equal to \(a_{n}-a_{m}\) by expanding and simplifying the sum. This demonstrates that the given sum is a telescoping sum.

Key Concepts

Summation Notation

Understanding summation notation is like learning a new language in mathematics – a shorthand for expressing the addition of a sequence of terms. It involves the sigma symbol \( \Sigma \) and can be intimidating at first glance, but it's essential for compactly writing out long sums.

Here's a quick breakdown: the sigma symbol \( \Sigma \) indicates summation, the index of summation (often \( i \) or \( n \) ) tells you where to begin, and the upper limit indicates where to end. For instance, \( \sum_{i=1}^{n} a_i \) means you sum all terms \( a_i \) from \( a_1 \) to \( a_n \) inclusively. To master telescope sums and many other series, getting familiar with this notation is pivotal.

In the context of a telescoping sum, the notation \( \sum_{i=m+1}^{n}\left(a_{i}-a_{i-1}\right) \) effectively tells us to add up the difference between consecutive sequence terms from \( a_{m+1} \) to \( a_n \) .

Sequence Differences

Closely related to summation is the concept of sequence differences. This is the essence of a telescoping series and is a useful tool for simplifying complex series. Think of it as looking at the steps between consecutive terms in a sequence.

When you're given a series where each term is a difference of consecutive elements, like \( a_{i} - a_{i-1} \), what you're dealing with is a sequence that tracks how much each term grows or shrinks from the previous one. So, it's essentially the 'step size' between terms in a sequence.

Series Simplification

When you're faced with long, intimidating series, the goal is to simplify. And telescoping series are a perfect example of how a complex series can collapse like a telescope into a simple expression.

As shown in the textbook solution, when a telescoping sum is expanded, it magically simplifies because intermediate terms cancel each other out. The only survivors are the first negative term and the last positive term. Observing these cancellations and rewriting the series in this reduced form is a critical skill in series simplification. It's like clearing the clutter to reveal the true nature of the series.

The simplification doesn't just make the expression cleaner, it often reveals the deeper relationships between the terms, which is especially important in fields like number theory and analysis.

Discrete Mathematics

Finally, this entire discussion falls under the broad and fascinating umbrella of discrete mathematics. But what is discrete math? It's all about structures that are countable or distinct and not continuous. Think integers, graphs, and logical statements — in contrast to real numbers which are uncountably infinite.

In the world of discrete mathematics, we are often interested in sequences (lists of discrete elements) and series (sums of sequences). What makes it so exciting is how it applies to computer science, cryptography, and other fields where the discrete nature of information comes into play.

Telescoping sums are a fantastic example of discrete mathematics in action. They show us how patterns emerge from sequences and can lead to simplified forms, illustrating the power of discrete structures to solve complex problems.