Question

Explain why the sequence of partial sums for a series with positive terms is an increasing sequence.

Short answer

Answer: A sequence of partial sums for a series with positive terms is an increasing sequence because each term of the sequence is greater than the previous term, due to the fact that the terms of the original series are all positive.

Step-by-step solution

Define a sequence of partial sums

A sequence of partial sums for a given series, say S, with terms {a_1, a_2, a_3, ...}, is a series formed by adding up consecutive terms of S, creating a new sequence {s_1, s_2, s_3, ...}, where: s_1 = a_1, s_2 = a_1 + a_2, s_3 = a_1 + a_2 + a_3, and so on.

Show that the terms of the partial sum sequence are increasing

Since all the terms of the original series S are positive (as stated in the exercise), this means that: a_1 > 0, a_2 > 0, a_3 > 0, and so on. Now, let's compare two consecutive partial sums, s_n and s_{n+1}: s_n = a_1 + a_2 + ... + a_n, s_{n+1} = a_1 + a_2 + ... + a_n + a_{n+1}. Since a_{n+1} > 0 (because all terms of the series S are positive), we can say that: s_{n+1} > s_n. This inequality shows that each term of the sequence of partial sums is greater than the previous term, meaning that the sequence of partial sums is an increasing sequence.

Conclusion

A sequence of partial sums for a series with positive terms is an increasing sequence because each term of the sequence is greater than the previous term, due to the fact that the terms of the original series are all positive.

Key concepts

Sequences in Calculus

In calculus, sequences are ordered lists of numbers following a specific rule. They are denoted as \(a_n\), where \(n\) represents the position in the sequence, and \(a_n\) is the value of the sequence at that position.

One fundamental type of sequence is the sequence of partial sums. When we talk about partial sums, we're describing the process of adding the terms of a series up to a certain point. This is important because it allows us to analyze the behavior of series over time, especially when dealing with infinite series.

The concept of partial sums is crucial for understanding the overall behavior of a series, because it forms the basis for determining whether the series converges or diverges—key concepts in the study of sequences and series in calculus.

Series Convergence

The concept of series convergence is central to understanding how sequences behave as their terms are added indefinitely. A series converges if the sequence of its partial sums approaches a finite number as more and more terms are added. In other words, as \(n\) becomes larger, the partial sums \(s_n\) get closer and closer to a specific value.

A simple way to test convergence for a series with positive terms is to examine whether its sequence of partial sums is bounded above. If there is a ceiling value that the sequence does not surpass, we can say that the series converges. Conversely, if the sequence of partial sums grows without bound, the series diverges. Understanding convergence is vital for many real-world applications, such as calculating compound interest or analyzing the behavior of a bouncing ball over time.

Properties of Series

Series possess several important properties that help us analyze and work with them effectively. One such property is monotonicity, which describes how the sequence of partial sums behaves—whether it is strictly increasing, decreasing, or constant. For instance, when dealing with positive term series, the sequence of partial sums is always increasing because each additional term in the series is positive, hence adding to the previous sum.

Another essential property is the boundedness of series. A series is considered bounded if there is a real number that the sequence of partial sums does not exceed. This is directly related to the concept of convergence, as any series that is increasing and bounded is guaranteed to converge.

There are other properties like the Cauchy criterion for convergence, tests for absolute convergence, and rules for combining series that are all critically important for comprehending the complex behavior of series within calculus.