Question

Suppose that \(a_{n} \geq 0\) for \(n \geq m\) and \(\sum a_{n}=\infty .\) Prove: If \(N\) is an arbitrary integer \(\geq m\) and \(J\) is an arbitrary positive number, then \(\sum_{n=N}^{N+k} a_{n}>J\) for some positive integer \(k\).

Short answer

Question: Prove that if a series \(\sum a_n\) has non-negative terms \(a_n \geq 0\) for \(n \geq m\) and converges to infinity (\(\sum a_n = \infty\)), then for any arbitrary integer \(N \geq m\) and any arbitrary positive number \(J\), there exists some positive integer \(k\) such that \(\sum_{n=N}^{N+k} a_n > J\).

Step-by-step solution

Assume \(\sum a_n = \infty\).

Since the given series \(\sum a_n\) converges to infinity, it means that for any positive number \(J\), we can always find a certain number of terms from the series that will exceed this value \(J\).

Pick an arbitrary integer \(N \geq m\) and an arbitrary positive number \(J\).

Let \(N\) be an arbitrary integer \(\geq m\), and let \(J\) be an arbitrary positive number.

Find the partial sums up to and including \(N\).

Define the sequence of partial sums \(S_N = \sum_{n=m}^{N} a_n\). This sequence will also converge to infinity as \(N\) approaches infinity.

Locate the first partial sum greater than \(J\).

Since the sequence of partial sums converges to infinity, there must exist some integer \(N_1 \geq N\) such that \(S_{N_1} > J\). This is because \(S_N\) represents the sum of the series \(\sum a_n\) up to the \(N\)-th term, and we must be able to reach any arbitrary positive number \(J\) by including enough terms from the series.

Define the difference between consecutive partial sums.

Let \(k = N_1 - N\). Consider the sum of the terms \(a_n\) from \(N\) to \(N+k\). We have: $$ \sum_{n=N}^{N+k} a_n = S_{N_1} - S_{N-1}. $$

Prove that the sum of the terms \(a_n\) from \(N\) to \(N+k\) is greater than \(J\).

From step 4, we know that \(S_{N_1} > J\). Since \(a_n \geq 0\) for all \(n \geq m\), it follows that \(S_{N_1} \geq S_{N-1}\). Thus, $$ \sum_{n=N}^{N+k} a_n = S_{N_1} - S_{N-1} > J - S_{N-1} \geq J. $$ This proves that for any arbitrary integer \(N \geq m\) and any arbitrary positive number \(J\), there exists some positive integer \(k\) such that \(\sum_{n=N}^{N+k} a_n > J\).

Key concepts

Infinite Series

An infinite series is a sequence of numbers in which you continue to add term after term indefinitely. For example, the series of natural numbers 1, 2, 3, 4, ..., written as \(\sum_{n=1}^{\infty} n\), is an infinite series. In the context of real analysis, the discipline of mathematics that deals with the properties of real numbers and the functions defined on them, the focus is on whether infinite series converge to a specific value or diverge to infinity.

As seen in the exercise, when we are given a series \(\sum a_n\) with non-negative terms and told that it diverges to infinity, it implies that as we keep adding terms, the sum grows without bound. This property is particularly crucial when dealing with the notion of partial sums, where we look at the sum of the series up to a certain number of terms. The exercise demonstrates a practical application of the concept of divergence in infinite series.

Partial Sum

A partial sum in the context of an infinite series is the sum of the first 'n' terms. It is denoted as \( S_N = \sum_{n=m}^{N} a_n \) for some integers m and N, where 'm' is the starting index of the sum, and 'N' is the ending index. The sequence of these partial sums, as 'N' grows, reveals a lot about the behavior of the series itself.

In our specific problem, we use partial sums to find a threshold beyond which the sum of the series surpasses any arbitrary positive number 'J'. By exploring the concept of a partial sum, students can better grasp how each section of an infinite series contributes to its overall sum, and why, if an infinite series diverges, the sums can become larger than any preassigned number, as demonstrated by the successive steps in the solution.

Real Analysis

Real analysis is a branch of mathematical analysis dealing with the set of real numbers and real-valued functions. This field of study is fundamental to many areas of mathematics and underpins the rigorous treatment of the limits of sequences and series, continuity, differentiation, and integration. It is within this framework of real analysis that the exercise is situated.

The exercise leverages the principles of real analysis to prove a property related to divergent infinite series, thereby showcasing real analysis's importance in understanding and manipulating infinite processes. Recognizing the divergence of a series and dealing with properties of series, such as the one described in the exercise, are central skills in real analysis and critical for higher mathematical education. In addressing such problems, students solidify their comprehension of abstract mathematical concepts by applying them to concrete examples.