Problem 115 - Essential Calculus

Question

Show that the series \(\sum_{n=1}^{\infty} a_{n}\) can be written in the telescoping form \(\sum_{n=1}^{\infty}\left[\left(c-S_{n-1}\right)-\left(c-S_{n}\right)\right]\) where \(S_{0}=0\) and \(S_{n}\) is the \(n\) th partial sum.

Step-by-Step Solution

Answer

The series can be written in telescoping form because, in the process of addition (series), each term \(S_{n-1}\) cancels the previous term \(-S_{n-1}\). As a result, we're left with the original series, demonstrating that it can be written in such a manner.

Step 1: Recall the Concept of Partial Sum

It is critical to initially remember the idea of a partial sum. In a given series, the ith partial sum \(S_{i}\) is the sum of the first i terms. In this way, by this definition, \(S_{0}\) ought to be 0 as it is the 'sum' of zero terms and each successive \(S_{i}\) is identified by the sum of \(S_{i-1}\) (the (i-1)th partial sum) and \(a_{i}\) (the ith term of the series). So we have: \(S_{i} = S_{i-1} + a_{i}\).

Step 2: Rewrite the Series in the telescoping form

First, let's draw attention to the telescoping series. Here we are dealing with \[\sum_{n=1}^{\infty}\left[\left(c-S_{n-1}\right)-\left(c-S_{n}\right)\right]\] If we distribute the subtraction sign in the brackets, this becomes \[\sum_{n=1}^{\infty}(c - S_{n-1} - c + S_{n})\] The two 'c' values cancel out and now we are left with \[\sum_{n=1}^{\infty}(S_{n} - S_{n-1})\]

Step 3: Relate the telescoping form to the original series

It’s visible that the terms now follow a pattern wherein every term \(S_{n-1}\) cancels the previous term \(-S_{n-1}\) for n > 1. So, when we sum everything, all these terms cancel out and we are left with the terms from the sequence that could not cancel out, which are \(S_{n}\) terms and the first \(S_{n-1}\) term, which equals 0. So what’s left is \(\sum_{n=1}^{\infty} a_{n}\). Hence, the proof.

Key Concepts

Partial Sums

In the context of an infinite series, the concept of partial sums helps us understand how the series builds up term by term. Imagine you have a series defined by its terms \(a_1, a_2, a_3, \ldots\), and you want to find the total sum when all terms are added. However, since the series is infinite, instead of aiming to solve the entire series at once, we look at parts of it.
The \(n\)th partial sum, denoted \(S_n\), is the sum of the first \(n\) terms of the series. Mathematically, it is expressed as:

• \( S_n = a_1 + a_2 + a_3 + \cdots + a_n \)

Partial sums are crucial because they allow us to study convergence. If the sequence of partial sums \(S_1, S_2, S_3, \ldots\) approaches a specific number as \(n\) goes to infinity, the series is said to converge. For example:

• \( S_0 = 0 \) because there are no terms to add initially.
• \( S_1 = a_1\)
• \( S_2 = S_1 + a_2 = a_1 + a_2 \)

This process continues, allowing each partial sum to include one additional term of the series at each step.

Infinite Series

An infinite series is formed from the sum of an infinite list of numbers, often represented as \( \sum_{n=1}^{\infty} a_n \). The core idea is to find if these sums lead to a finite value or diverge. Sometimes, these sums may not settle on a precise number, making them divergent.
Consider the classic series arrangement:

• Geometric Series: \(a + ar + ar^2 + \cdots\)
• Harmonic Series: \(1 + \frac{1}{2} + \frac{1}{3} + \cdots\)

To better understand an infinite series, we often investigate its partial sums. By evaluating these progressively larger portions, we can see how the series behaves over a long term.
The behavior of partial sums determines whether a series is convergent or divergent:

• If the series converges, its partial sums approach a fixed value.
• If the series is divergent, the sums either increase indefinitely or oscillate without nearing any fixed point.

Identifying whether an infinite series is convergent or divergent is crucial, as it opens up various implications for mathematical calculations and analyses.

Series Manipulation

Manipulating series involves rewriting or reformatting them to uncover hidden properties or simplify calculations. A telescope series is one innovative way to express a series. It revolves around structuring a series so that most terms cancel when summed.
This exercise demonstrated that the series \( \sum_{n=1}^{\infty} a_n \) can be restructured into a telescoping form. Here’s how it's broken down:

• The series is rewritten in the form: \( \sum_{n=1}^{\infty}\left[(c-S_{n-1})-(c-S_{n})\right] \), where \(S_n\) is the \(n\)th partial sum.
• Distributing terms results in cancellation: \(\sum_{n=1}^{\infty} (S_n - S_{n-1})\).

The remarkable outcome of this manipulation is that when summed, most terms in the series annihilate each other, leaving behind only terms that don't cancel. These terms are crucial in calculating the sum or reaching the central series form, often turning something complex into something more concise.
Understanding series manipulation tools like this enable one to uncover simplifications in problems involving complex or infinite series. It makes solving such problems practical and, often, insightful in recognizing patterns.