Question

Prove that the convergence or divergence (pointwise or uniformly) of a sequence \(\left\\{f_{m}\right\\}\), or a series \(\sum f_{m}\), of functions is not affected by deleting or adding a finite number of terms. Prove also that \(\lim _{m \rightarrow \infty} f_{m}\) (if any) remains the same, but \(\sum_{m=1}^{\infty} f_{m}\) is altered by the difference between the added and deleted terms.

Short answer

Convergence is not affected by finite alterations; limits remain the same, series sums change by the finite alteration difference.

Step-by-step solution

Introduction to Problem

We need to show that adding or removing a finite number of terms in a sequence or series of functions does not affect their convergence or divergence. We also need to demonstrate the effects on their limits and sums.

Consider a Sequence or Series

Let's consider a sequence of functions \(\{f_m\}\) and a series \(\sum f_m\). Our goal is to prove that the convergence characteristics do not change when a finite number of these functions are altered.

Analyze Pointwise Convergence

Pointwise convergence means that for each point \(x\), the sequence \(\{f_m(x)\}\) converges. Altering a finite number of terms adds or subtracts finitely many values from this sequence. Since convergence depends on \(m\rightarrow \infty\), finite changes do not affect this.

Analyze Uniform Convergence

For uniform convergence, \(f_m\) converges uniformly to \(f\) if \(\sup_x |f_m(x) - f(x)| \to 0\) as \(m\) approaches infinity. Adding or deleting finite terms results in finite changes, hence the eventual distances between \(f_m\) and \(f\) at each \(x\) remain unchanged for large \(m\).

Consider Limits and Alterations

The limit \(\lim_{m \to \infty} f_m\) is determined by the behavior as \(m\) becomes very large. The finite number of alterations does not change the sequence's behavior for large \(m\), hence the limit remains the same.

Analyze Series Sum Alterations

For the series, the sum \(\sum_{m=1}^\infty f_m\) is indeed altered by the sum of the deleted terms minus those added. The altered terms are only finite, so their contribution is finite.

Key concepts

Pointwise Convergence

Pointwise convergence is a straightforward concept in the analysis of function sequences. Imagine we have a sequence of functions \(\{f_m(x)\}\) that depend on the variable \(x\). We say that this sequence converges pointwise to a function \(f(x)\) if, for every fixed \(x\), the sequence \(\{f_m(x)\}\) approaches \(f(x)\) as \(m\) approaches infinity.
The fundamental characteristic of pointwise convergence is that it considers convergence at each individual point separately. Therefore, even if the values at a finite number of points are altered, the overall pointwise convergence across the domain is unaffected.
For example:

• If we have \(f_m(x) = \frac{x}{m}\), for each fixed \(x\), as \(m\) tends to infinity, \(f_m(x)\) will converge to 0, regardless of any finite changes made to the sequence.
• Only the behavior at an infinite number of points matters, ensuring the limit function remains intact.

Uniform Convergence

Uniform convergence deals with a stronger type of convergence compared to pointwise. For uniform convergence, a sequence of functions \(\{f_m\}\) approaches a function \(f(x)\) uniformly if, regardless of the value of \(x\), the difference \(\sup_x |f_m(x) - f(x)|\) can be made as small as desired, uniformly over the entire domain, as \(m\) becomes very large.
This means that the entire sequence gets closer to \(f(x)\) in such a way that for all \(x\) in its domain, the distances between \(f_m(x)\) and \(f(x)\) remain small simultaneously for large \(m\).
Key points to remember:

• Finite changes to the sequence do not affect uniform convergence because they don't impact the behavior as \(m\) tends to infinity.
• Since convergence is uniform, we maintain the same bounded control over the entire sequence post alteration, ensuring unchanged final convergence.

Function Series

A function series arises when we sum a sequence of functions \(\{f_m\}\). The series is written as \(\sum_{m=1}^{\infty} f_m(x)\). Just like number series, the convergence of a function series depends on whether the infinite sum results in a finite limit for each \(x\).
If you alter terms by adding or removing a few, it directly changes only a finite part of the sum, leaving the infinite tail unchanged.
When considering alterations:

• The series convergence remains consistent because only infinitely many terms drive the final convergence.
• However, the value \(\sum f_m\) does get adjusted by the difference between any terms added and deleted since these only amount to a finite change.

In simpler terms, the sequence of partial sums that determine series convergence overlooks any finite disruptions, as they become negligible with respect to the total sum.