Question

Prove: If \(\lim _{i \rightarrow \infty} a_{j}^{(i)}=a_{j}(j \geq 1)\) and \(\left|a_{j}^{(i)}\right| \leq \sigma_{j}(i, j \geq 1),\) where \(\sum_{j=1}^{\infty} \sigma_{j}<\) \(\infty,\) then \(\lim _{i \rightarrow \infty} \sum_{j=1}^{\infty} a_{j}^{(i)}=\sum_{j=1}^{\infty} a_{j}\)

Short answer

Question: Prove that if the limit of each term in a series of sequences converges, and the absolute value of each term is bounded by another series that converges, then the limit of the sum of terms of each sequence also converges to the sum of the limits of the terms. Solution: We used the Weierstrass M-test to show that the series of the sequences converges uniformly, and then we used the property that the limit of a uniformly convergent series equals the series of the limits of the terms. By doing so, we proved that the limit of the sum of terms of the sequences is equal to the sum of the limits of the terms.

Step-by-step solution

Define the convergence of the sequences

Given that \(\lim_{i\to\infty} a_j^{(i)} = a_j\) for all \(j \geq 1\), it means as \(i\) goes to infinity, the terms of each sequence \(a_j^{(i)}\) converge to the corresponding terms of the limiting sum \(a_j\).

Define the given inequality

We are given that \(\left|a_{j}^{(i)}\right| \leq \sigma_{j}\) for all \(i, j \geq 1\). This inequality represents the fact that the absolute value of each term in the sequences is bounded by the terms of the convergent series \(\sum_{j=1}^{\infty} \sigma_{j}\).

Apply Weierstrass M-test

We can apply Weierstrass M-test to this problem. The Weierstrass M-test states that if \(|f_j(x)| \leq M_j\) for all \(x \in A\), and \(\sum_{j=1}^{\infty} M_j\) converges, then \(\sum_{j=1}^{\infty} f_j(x)\) converges uniformly on \(A\). In our case, we have \(f_j(i) = a_j^{(i)}\), \(M_j = \sigma_j\), and \(A ={1,2,\ldots}\). We know that \(\sum_{j=1}^{\infty} \sigma_j\) converges, so by the Weierstrass M-test, we have that \(\sum_{j=1}^{\infty} a_j^{(i)}\) converges uniformly on \(i\).

Use uniform convergence properties

Since the series \(\sum_{j=1}^{\infty} a_j^{(i)}\) converges uniformly on \(i\), we can use the property that the limit of a uniformly convergent series equals the series of the limits of the terms: $$\lim_{i\to\infty} \sum_{j=1}^{\infty} a_j^{(i)} = \sum_{j=1}^{\infty} \lim_{i\to\infty} a_j^{(i)}$$

Substitute given limit

We know that \(\lim_{i\to\infty} a_j^{(i)} = a_j\), so we can substitute this into the equation from Step 4: $$\lim_{i\to\infty} \sum_{j=1}^{\infty} a_j^{(i)} = \sum_{j=1}^{\infty} a_j$$

Conclusion

We have shown that the limit of the sum of terms of the sequences is equal to the sum of the limits of the terms, meaning that \(\lim_{i\to\infty} \sum_{j=1}^{\infty} a_j^{(i)} = \sum_{j=1}^{\infty} a_j\). This completes the proof.

Key concepts

Weierstrass M-test

The Weierstrass M-test is a powerful tool in real analysis for determining the uniform convergence of infinite series consisting of functions. It establishes criteria when a series of functions converges uniformly. Here's a more digestible break down:

Imagine you have a sequence of functions, and each function is 'capped' or bounded by a certain value, much like putting a leash on a dog so it can't wander too far. Now, if the series of these 'caps' — the leash lengths, if you will — adds up to a finite total length, then no matter how many functions you include in your series, their 'leashed' values won't skip off to infinity together. The result is that your function series behaves nicely and converges uniformly. This concept can seem daunting at first, but it's akin to ensuring that each member of a choir can't sing louder than a fixed volume; if they all follow this rule, the choir's collective singing won't get unbearably loud, no matter how many singers join.

When applied to the problem at hand, the Weierstrass M-test essentially guarantees that the series of sequences \(a_j^{(i)}\) has a well-mannered sum, and this allows us to safely exchange the order of taking limits and summing the series.

Uniform Convergence

Moving on to another fundamental concept, uniform convergence is akin to an entire flock of birds landing together in perfect synchronization — it's about all the parts of a series settling down to their limits in tandem, at the same rate.

In the context of functions or series of sequences, uniform convergence ensures that for any small distance you choose, there's a stage beyond which the entire function (or sequence), not just the individual terms, stays within that distance from its ultimate shape — its limit. It's this attribute that allows for goodies like the ability to swap limits and series (integration or differentiation, too), and it’s essential for ensuring that many properties of the sequence carry over to its limit.

The given exercise takes advantage of these features. Since the series converges uniformly, we have confidence that the limit is well-behaved and that no term in the series causes mischief by diverging away.

Series of Sequences

Think of a series of sequences like a layer cake. Each slice of cake represents a sequence, and the whole cake is the series. In the exercise, each sequence is defined by \(a_j^{(i)}\), which are like the individual slices that when combined (summed up) layer by layer, give us the whole dessert — the series.

What makes these series intriguing is how they converge. If each layer, no matter how deep into the cake, is moving closer to a determined ideal slice (the limit), then the series can converge to a predictable pattern. This behavior is what's being tested when looking at the convergence of a series of sequences; we're checking if each successive layer of the cake converges to what we would expect in an ideal cake, layer by layer.

Understanding the series of sequences helps in analyzing complex functions and solving advanced problems in various fields of mathematics and engineering.

Real Analysis

Lastly, real analysis is a branch of mathematics dealing with real numbers and the real-valued functions of real variables. It digs deep into the concepts of limits, series, continuity, and more.

In layman’s terms, real analysis is about getting to the heart of what happens in the infinite and infinitesimal scales of numbers and their behaviors. It tasks itself with formalizing the intuition behind numbers growing very large or shrinking very small, and series that go on forever towards an intended target. It's a bit like being a number detective, where you figure out not just if the values reach their destination but how they go about it.

Understanding all of this sets the stage for higher mathematics and allows us to rigorously explore and solve problems in calculus, differential equations, and more. It’s the foundation upon which much of mathematical analysis is built.