Question

Prove the following two theorems: The Weierstrass Criterion \(^{3} .\) If a series \(\sum_{i=1}^{\infty} f_{i}\) of functions \(f_{i}: X \rightarrow M\) is majorized by a convergent numerical series $$ \left\|f_{i}\right\| \leq a_{i}, \quad \sum_{i=1}^{\infty} a_{i}<\infty, \quad a_{i} \in \boldsymbol{R} $$ then it converges absolutely and uniformly on \(X\).

Short answer

Question: Prove the Weierstrass Criterion using the definitions of absolute and uniform convergence. Answer: According to the Weierstrass Criterion, if we have a series of functions \(f_i: X \rightarrow M\) and a convergent numerical series \(a_i\), such that \(\left\|f_i\right\| \leq a_i\) for all \(i\), then the series of functions converges absolutely and uniformly on \(X\). By applying the Weierstrass M-test, we show that the given conditions satisfy the test with \(M_i = a_i\), which implies the series \(\sum_{i=1}^{\infty} f_i(x)\) converges absolutely and uniformly on \(X\).

Step-by-step solution

Define absolute and uniform convergence of a series

Absolute convergence of a series \(\sum_{i=1}^{\infty} f_{i}\) means that the series of absolute values of its terms, \(\sum_{i=1}^{\infty} |f_{i}|\) converges. Uniform convergence of a series of functions \(\sum_{i=1}^{\infty} f_{i}\) means that for any \(\epsilon > 0\), there exists an integer \(N\) such that for all \(x \in X\) and all \(n \geq N\) we have \(\left|\sum_{i=n}^{\infty} f_{i}(x)\right| < \epsilon\). We will now proceed to prove the Weierstrass Criterion using these definitions.

Apply the Weierstrass M-test

To show that the series of functions \(f_i\) converges absolutely and uniformly, we can use the Weierstrass M-test. It states that if there exists a sequence of positive constants \(M_i\) such that \(|f_i(x)| \le M_i\) for all \(i\) and all \(x \in X\), and \(\sum_{i=1}^{\infty} M_i\) converges, then the series \(\sum_{i=1}^{\infty} f_i(x)\) converges absolutely and uniformly on \(X\).

Show that the given conditions satisfy the Weierstrass M-test

We are given that \(\left\|f_i\right\| \leq a_i\) and \(\sum_{i=1}^{\infty} a_i < \infty\), which means that we have a convergent numerical series. By definition of the norm, we have \(\left\|f_i\right\| = \sup_{x \in X} |f_i(x)|\). Therefore, \(|f_i(x)| \leq a_i\) for all \(x \in X\) and all \(i\). This inequality satisfies the condition for the Weierstrass M-test, with \(M_i = a_i\).

Conclude the proof

Since the given conditions satisfy the Weierstrass M-test, the series \(\sum_{i=1}^{\infty} f_i(x)\) converges absolutely and uniformly on \(X\) according to the test. This completes the proof of the Weierstrass Criterion.

Key concepts

Absolute Convergence

Absolute convergence is a crucial concept when analyzing series of functions. When we mention that a series

• \(\sum_{i=1}^{\infty} f_{i}\)

converges absolutely, it means that the series formed by taking the absolute values of each function term,

• \(\sum_{i=1}^{\infty} |f_{i}|\)

also converges. This is a stronger form of convergence and assures that the series of original functions will sum to a finite number. Being absolutely convergent implies some very stable mathematical properties that are desirable. Absolute convergence is significant because if a series converges absolutely, it ensures convergence regardless of the order in which terms are added. This is particularly helpful in mathematical function analysis as it guarantees consistent results even under rearrangements of terms.

Uniform Convergence

While absolute convergence deals with the sum of absolute values, uniform convergence tells us about the behavior of a function series across an entire domain. For a series of functions \(\sum_{i=1}^{\infty} f_{i}\) to be uniformly convergent, for any small positive number (\(\epsilon > 0\)), there must exist a natural number \(N\) such that for all \x \ in \the set \\(X\) and all \(n \geq N\), the inequality \(\left|\sum_{i=n}^{\infty} f_{i}(x)\right| < \epsilon\)\holds true.

This definition might look similar to pointwise convergence, but there is a pivotal difference. Uniform convergence manages the approximation quality uniformly over the entire domain. It ensures that the partial sums \(\sum_{i=1}^{n} f_{i}(x)\)approximate a limit function alike for every single point \(x\). This is powerful because it allows us to exchange limits and integrals freely, making calculations and theoretical advancements easier and more robust.

Weierstrass M-test

The Weierstrass M-test is an Elegant tool to determine the convergence of a series of functions. It assures us about both absolute and uniform convergence under a given set of conditions. According to the Weierstrass M-test, suppose for each function term \(f_i\), there exists a positive constant \(M_i\) such that for all \(x \in X\),

• \(|f_i(x)| \le M_i\)

if this sequence \(\sum_{i=1}^{\infty} M_i\) converges, then the function series \(\sum_{i=1}^{\infty} f_i(x)\) converges absolutely and uniformly on the domain \(X\).

This test is particularly helpful as it requires us only to work with numerical bounds \(M_i\), rather than more complex function behaviors. Not only does it simplify verifying convergence, but it also directly connects to the original problem's condition—where the series of norms \(a_i\) are given as convergent. By leveraging the Weierstrass M-test, we can conclude with certainty about the series convergence properties over its entire domain, bringing both theoretical and practical ease to mathematical analysis and solutions.