Problem 11 - Basic Analysis: Introduction to Real Analysis

Question

a) If \(\sum a_{n}\) is convergent to a (in the usual sense), show that \(\sum a_{n}\) is Cesàro summable (see above) to a. b) Show that in the sense of Cesàro \(\sum(-1)^{n}\) is summable to \(1 / 2\). c) Let \(a_{n}:=k\) when \(n=k^{3}\) for some \(k \in \mathbb{N}, a_{n}:=-k\) when \(n=k^{3}+1\) for some \(k \in \mathbb{N},\) otherwise let \(a_{n}:=0 .\) Show that \(\sum a_{n}\) diverges in the usual sense, (partial sums are unbounded), but it is Cesàro summable to 0 (seems a little paradoxical at first sight).

Step-by-Step Solution

Answer

a) Convergent \(\sum a_n\) is also Cesàro summable to \(a\). b) \(\sum (-1)^n\) is Cesàro summable to \(1/2\). c) \(\sum a_n\) diverges but is Cesàro summable to 0.

Step 1: Recognizing Convergence in the Usual Sense

Given that the series \(\sum a_n\) converges normally to \(a\), this means that its partial sums \(s_N = a_1 + a_2 + \ldots + a_N\) converge to \(a\) as \(N \to \infty\). Mathematically, this is expressed as \(\lim_{N \to \infty} s_N = a\).

Step 2: Understanding Cesàro Summability

A series is Cesàro summable to \(S\) if \(\lim_{N \to \infty} \frac{1}{N} \sum_{k=1}^N s_k = S\), where \(s_k\) are the partial sums of the original series. Our goal is to show that if \(\sum a_n\) converges to \(a\), then it is also Cesàro summable to \(a\).

Step 3: Proving Cesàro Summability for Part (a)

Because \(\sum a_n\) is convergent, for each \(\varepsilon > 0\), there exists \(M\) such that for all \(n > M\), \(|s_n - a| < \varepsilon\). To show Cesàro summability, consider the sequence of arithmetic means, \(\frac{1}{N}\sum_{k=1}^{N}s_k\). By the Stolz–Cesàro theorem or manipulating the arithmetic mean, since each \(s_k \to a\), we have \(\frac{1}{N}\sum_{k=1}^{N}s_k \to a\), proving Cesàro summability to \(a\).

Step 4: Understanding Alternating Series for Part (b)

\(\sum (-1)^n\) is the series \(1 - 1 + 1 - 1 + \ldots\). Its partial sums oscillate between 1 and 0. Observe that the average of the first \(N\) terms, \(\frac{1}{N}\sum_{k=1}^{N}s_k\), approaches \(\frac{1}{2}\) as \(N\) goes to infinity.

Step 5: Proving Cesàro Summability for \((-1)^n\)

Note that for even \(N\), the partial sum \(s_N = 0\) and for odd \(N\), \(s_N = 1\). As \(N\) increases, the average \(\frac{1}{N}\sum_{k=1}^{N}s_k\) reaches \(\frac{1}{2}\) because the contributions from both even and odd sums ( half converge to 0 and half to 1) balance out, showing it is Cesàro summable to \(\frac{1}{2}\).

Step 6: Defining the Series in Part (c)

The sequence is defined as follows: \(a_n = k\) when \(n = k^3\), \(a_n = -k\) when \(n = k^3 + 1\), and \(a_n = 0\) otherwise. This means there are large intervals of zeros punctuated by spikes of positive and negative numbers.

Step 7: Proving Divergence in Usual Sense for Part (c)

Analyze the series' convergence by considering partial sums. If multiple large terms \(+k\) at \(k^3\) and \(-k\) at \(k^3+1\) are observed while intervening terms are zeros, the partial sums do not stabilize. As \(k\) grows, the sums can grow infinitely large and unbounded, showing regular divergence.

Step 8: Cesàro Summability Analysis for Part (c)

To prove Cesàro summability, the main idea is to account for the contribution of terms in long sequences of zeros. When averaged out, for each \(k\), the impact of \(+k\) and \(-k\) is negligible over increasing \(N\), thus the Cesàro mean \(\frac{1}{N}\sum_{k=1}^{N}s_k\) tends to 0 as the zeros outweigh spikes, proving Cesàro summability to 0.

Key Concepts

Convergent Series

In mathematics, a series is a sum of terms from a sequence. A series is said to be convergent if the sequence of its partial sums approaches a specific value as the number of terms increases to infinity. When a series converges, it means that you can keep adding more and more terms, yet the total sum won't stray beyond a particular number.
For example, consider the series formed by adding fractions that get progressively smaller:

• \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots \)

This series converges to 2. Even though we keep adding more terms, they become so tiny that their overall impact diminishes, allowing the series to settle around 2.
This concept is crucial in calculus and analysis, as it allows us to make sense of infinite sums, like the ones seen in power series or Fourier series. A convergent series can also be Cesàro summable, which is another way of exploring how a sequence can approach a limit.

Alternating Series

An alternating series is one where the terms switch signs between positive and negative. This back-and-forth nature can lead to an interesting phenomenon where the series converges despite its inherent oscillation. The classic example is:

• \(1 - 1 + 1 - 1 + \cdots\)

This series is Cesàro summable, meaning that by examining an average of the partial sums, we attain a convergence value, in this case, 0.5. With alternating series, the party trick is that error bounds can often be derived directly from the terms themselves.
This is captured in the Alternating Series Test, which provides conditions under which these types of series will converge. Namely, if the absolute value of the terms decreases over time, and approaches zero, then the series converges.

Partial Sums

Partial sums are essentially the building blocks of a series. For any series \(\sum a_n\), the \(N\)-th partial sum is the sum of the first \(N\) terms:

• \(s_N = a_1 + a_2 + \cdots + a_N\).

Partial sums are critical because they allow us to investigate whether a series converges, diverges, or is Cesàro summable. By examining partial sums, we can better understand the behavior of infinite series.
They provide snapshots that can signify stability or an endless increase, as seen in divergent series. Taking averages of these sums, as is done in Cesàro summation, offers another layer to their analysis, smoothing out wild oscillations that might otherwise obscure a series' convergence.

Divergence in Series

Not all series converge; some, known as divergent series, do not settle on a particular value as more and more terms are added. In a divergent series, partial sums can grow indefinitely, or oscillate without approaching a specific number. This makes conventional summation methods futile.
Consider a series like:

• \(1 + 1 + 1 + \cdots\)

Here, no matter how many terms you sum, the total continues to increase, resulting in divergence.
However, divergence does not necessarily mean a series is meaningless. Tools like Cesàro summation offer ways to explore the characteristics of divergent series by averaging partial sums. Thus, while a series may not converge in the traditional sense, there can still be value in examining its behavior through alternative means, as shown in Cesàro summability with divergent patterns.