Explain Convergence Theorem & Contradicting Statements

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SUMMARY

The discussion centers on the Convergence Theorem and the apparent contradictions in its statements regarding series convergence. The user highlights confusion over the assertion that if the ratio of terms is equal to unity, the series diverges, while also suggesting that the series may converge based on subsequent terms. The key takeaway is that the series defined by u1 + u2 + u3 + ... + un can diverge or converge depending on the behavior of the succeeding terms, specifically un+1, un+2, etc. This indicates a nuanced understanding of convergence that requires careful analysis of the series' terms.

PREREQUISITES
  • Understanding of the Convergence Theorem in mathematical analysis
  • Familiarity with infinite series and their properties
  • Knowledge of the concept of term ratios in sequences
  • Basic principles of mathematical divergence and convergence
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  • Study the detailed proofs of the Convergence Theorem in mathematical analysis
  • Learn about the behavior of series with constant ratios, particularly in the context of geometric series
  • Explore examples of divergent and convergent series to solidify understanding
  • Investigate the implications of the ratio test and its application in series convergence
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Mathematicians, students of calculus, and anyone interested in advanced mathematical concepts related to series convergence and divergence.

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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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 + ...
 

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