How can i prove these two convergence theorem?

In summary, if an and bn are convergent, then their sum and product series are also convergent, and if there exists a constant k>0 such that |bn| > k for all n=1, 2, ..., then the quotient series an/bn is also convergent. This can be proven by showing that the limits of the sum and product series are the same as the postulated limit, and using the definition of convergence to find Na and Nb such that |a_n - a| < \epsilon and |b_n - b| < \epsilon.
  • #1
1. If {an} and {bn} are convergent, then {an士bn} and {anbn} are also convergent

2. If {an} and {bn} are convergent and there exists a constant k>0 such that |bn| > k for all n=1, 2, ..., then {an/bn} is also convergen
 
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  • #2
Well, given that an converges to a and bn converges to b, you could "guess" the limits of the other series and then try to prove that they actually are the limits from the definition.

E.g. you can try to show that
[tex]\forall_{\epsilon > 0} \exists_{N_+} : n > N_+ \implies |(a_n + b_n) - L| < \epsilon[/tex]
where L = a + b is the postulated limit. Of course you already know that given such an [itex]\epsilon[/itex] you can find Na and Nb such that
[tex]n > N_a \implies |a_n - a| < \epsilon[/tex]
and
[tex]n > N_b \implies |b_n - b| < \epsilon[/tex]
 

1. What is a convergence theorem?

A convergence theorem is a mathematical result that states that certain sequences or series will approach a specific limit or value as the number of terms in the sequence or series increases. It is used to prove the convergence of mathematical expressions and can be applied to various fields such as calculus, statistics, and physics.

2. How can I prove a convergence theorem?

To prove a convergence theorem, you need to show that the sequence or series under consideration satisfies the conditions of the theorem. This can involve using mathematical techniques such as limit theorems, comparison tests, or the Cauchy criterion. It is important to carefully examine the assumptions of the theorem and provide a rigorous proof using logical reasoning and mathematical techniques.

3. What are some common examples of convergence theorems?

Some common examples of convergence theorems include the Monotone Convergence Theorem, which states that a monotone sequence that is bounded will converge to a limit, and the Root Test, which is used to determine the convergence of infinite series. Other examples include the Ratio Test, the Comparison Test, and the Integral Test.

4. Why are convergence theorems important in science?

Convergence theorems are important in science because they provide a powerful tool for analyzing and understanding the behavior of mathematical expressions. They allow scientists to determine the convergence or divergence of a sequence or series, which can have important implications in fields such as physics, engineering, and economics. Convergence theorems also help to establish the validity and accuracy of mathematical models and theories.

5. Are there any limitations to convergence theorems?

While convergence theorems are useful tools, they do have some limitations. In certain cases, the conditions of a convergence theorem may not be satisfied, making it difficult to determine the convergence of a sequence or series. Additionally, some convergence theorems only apply to specific types of sequences or series, so they may not be applicable to all mathematical expressions. It is important to carefully consider the assumptions and limitations of a convergence theorem before applying it to a problem.

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