Explain Convergence Theorem & Contradicting Statements

In summary, The conversation is discussing a mathematical concept related to infinite series and their convergence or divergence. Specifically, they are discussing a scenario in which the ratio of terms is equal to unity and how that affects the convergence of the series. The first highlighted portion states that the series will diverge, but the second highlighted portion suggests that it may or may not diverge depending on the proceeding terms. The conversation ends with a clarification on the series' actual terms and a confirmation of the correct understanding.
  • #1
Miike012
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Can someone explain to me what they are saying in the paint document? Because to me it seems like the statements are contradicting.

The first paragraph starts off with..." Let the fixed term be denoted..."

My concern is when the paragraph states.. "If the ratio is equal to unity, each of the succeeding terms is equal to u1 and the sum of n terms is equal to nu1hence the series is divergent.

First off that doesn't make sense and let me explain: Let the fixed term be denoted un and Let the infinite series be equal to
u1 + u2 + u3 + ... + un + ...

Now I will state what they said... If the ratio is equal to unity (un/un-1 = 1 or un = un-1) then each of the succeeding terms is equal to un and the sum of the n terms is equal to nun.

or in other words the series u1 + u2 + u3 + ... + un + ...
is equal to

nun + un+1 + un+2 + ...

However the first n terms which is equal to nun is a FINITE quantity and therefore the series will converge if the proceeding terms un+1 + un+2 + ... converge or diverge if the proceeding terms un+1 + un+2 + ... diverge.

Is this correct?

Another question:
Why do they say in the first highlighted portion that it will diverge but in the second highlighted portion say that it may or may not diverge?
 

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  • #2
Miike012 said:
or in other words the series u1 + u2 + u3 + ... + un + ...
is equal to

nun + un+1 + un+2 + ...
No, it's equal to u1 + u2 + u3 + ... + un + un + un + un + un + un + un + ...
 

1. What is the Convergence Theorem?

The Convergence Theorem, also known as the Limit Comparison Test, is a mathematical theorem that allows for the comparison of the convergence of two infinite series. It states that if the ratio of two series approaches a finite limit, then the two series will either both converge or both diverge.

2. How does the Convergence Theorem work?

The Convergence Theorem works by comparing the ratio of two infinite series. If the limit of the ratio is a finite number, then the two series have the same convergence behavior. If the limit is 0, then both series converge, and if the limit is infinity, then both series diverge.

3. Can the Convergence Theorem be used to prove a series converges?

Yes, the Convergence Theorem can be used to prove that a series converges. If the limit of the ratio between the series is a finite number, then the theorem guarantees that both series will converge.

4. What is a contradicting statement in mathematics?

A contradicting statement in mathematics is a statement that is logically opposite of another statement. In other words, if one statement is true, the contradicting statement must be false, and vice versa. In mathematical proofs, contradicting statements can be used to show that a certain assumption or hypothesis is not valid.

5. How can contradicting statements be used in the Convergence Theorem?

In the Convergence Theorem, contradicting statements can be used as a way to prove that a series does not converge. By showing that the ratio of two series does not approach a finite limit, it can be concluded that the two series have different convergence behavior, contradicting the theorem's assumption that they should have the same convergence behavior.

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