Question

Suppose that \(a_{n} \geq 0\) is a sequence of numbers. Explain why the sequence of partial sums of \(a_{n}\) is increasing.

Short answer

The sequence of partial sums is increasing because each term \(a_n\) is \(\geq 0\), so adding these non-negative terms keeps or increases the total.

Step-by-step solution

Identify Key Information

We start by considering the sequence of numbers \(a_n\), where each term \(a_n\) is non-negative, meaning \(a_n \geq 0\) for all \(n\). The sequence of partial sums is a related sequence formed by adding the terms \(a_n\) cumulatively.

Define the Sequence of Partial Sums

The sequence of partial sums \(S_n\) is defined as \(S_n = a_1 + a_2 + \, \ldots \, + a_n\). This sequence represents the cumulative total of the first \(n\) terms of \(a_n\).

Consider the General Term of the Partial Sum Sequence

To check if \((S_n)\) is increasing, we should demonstrate that \(S_{n+1} \geq S_n\) for all \(n\). The primary relationship is described by: \(S_{n+1} = S_n + a_{n+1}\).

Apply Properties of Non-Negative Numbers

Since \(a_{n+1} \geq 0\), the addition of a non-negative number \(a_{n+1}\) to \(S_n\) ensures that \(S_{n+1} \geq S_n\). This is because adding zero or a positive number to another number will always yield a result that is equal to or greater.

Conclude That the Sequence is Increasing

Hence, by the principle that adding a non-negative number maintains or increases a sum, the sequence of partial sums \(S_n\) is increasing. As \(n\) progresses, each new term \(a_{n+1}\) (non-negative) added to the previous total \(S_n\) results in an equal or larger sum.

Key concepts

Increasing Sequence

An increasing sequence is a series of numbers where each term is larger than or equal to the term before it. In mathematical terms, a sequence \((a_1, a_2, a_3, \ldots)\) is called increasing if for all \(n\), \(a_{n+1} \geq a_n\). This property is crucial in understanding the behavior of many mathematical and real-world phenomena. With increasing sequences, we always move forward or stay level, but never decrease.

In the context of the original exercise, the partial sums sequence \(S_n\) is increasing. This conclusion is because each new term added, \(a_{n+1}\), is non-negative. Thus, when you add it to the current sum \(S_n\), the new sum \(S_{n+1}\) is never smaller than \(S_n\). As you keep adding these non-negative terms, the sequence continues to grow or remain the same.

Non-negative Sequences

A non-negative sequence is a sequence where each element is zero or a positive number. No negative numbers are allowed in this type of sequence, hence the name. Formally, a sequence \((a_1, a_2, a_3, \ldots)\) is said to be non-negative if every term \(a_n \geq 0\).

This property ties neatly into our understanding of partial sums increasing. Since every term in our sequence is non-negative, when adding these terms to form a partial sum sequence \(S_n\), the sum can only stay the same or get larger. This non-negative restriction ensures that we are always either maintaining the current sum or adding to it, never subtracting or reducing it.

Cumulative Sums

Cumulative sums, often called partial sums, are sums formed by sequentially adding terms of a sequence. For instance, the cumulative sum sequence of \((a_1, a_2, a_3, \ldots)\) is \((S_1, S_2, S_3, \ldots)\), where each \(S_n = a_1 + a_2 + \ldots + a_n\).

These sums are incredibly useful, particularly in data analysis, to observe how a total builds over time. In our exercise scenario, we demonstrate that the cumulative sum sequence preserves the increasing property of the original sequence of terms. Every time we add the next term \(a_{n+1}\), provided that it is non-negative, the next cumulative sum value \(S_{n+1}\) will be equal to or greater than the previous \(S_n\). This growing sequence of cumulative sums reflects the accumulation of the individual non-negative terms from the original sequence.