How Can the Ratio Test Be Proven for Series Divergence When the Ratio Exceeds 1?

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    Proof Ratio Test
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Homework Help Overview

The discussion revolves around proving the Ratio Test for series divergence, particularly in cases where the ratio exceeds 1. Participants are exploring foundational concepts in series convergence and divergence.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are considering comparisons to known series, such as geometric series, to establish a basis for understanding divergence. Questions about how to demonstrate that terms of a series can exceed finite values are also raised.

Discussion Status

The discussion is in an exploratory phase, with some participants suggesting comparisons to geometric series and others proposing intuitive reasoning about the implications of a ratio greater than 1. No consensus has been reached, but various lines of reasoning are being examined.

Contextual Notes

There is a mention of standard theory related to the topic, and participants are encouraged to consider relevant examples, although specific details or examples are not provided in the discussion.

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Homework Statement
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Relevant Equations
Please see below
For (a) and (b),
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Does someone please know how to prove this? I don't have any ideas where to start.

Thanks!
 
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a) b) is standard theory.

Relevant examples also included in the link above.
 
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A good starting point is to compare it to a geometric series. For example if ##c=1/3## can you think of a series that converges whose terms are eventually guaranteed to be larger than the ##x_n##?
 
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If the ratio is > 1, then compare to adding a nonzero number to itself " infinitely often", show it will eventually surpass any finite value.
 
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