Is superlinear convergence always better than linear convergence?

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Homework Help Overview

The discussion revolves around the concepts of linear and superlinear convergence in sequences, exploring their definitions and implications. Participants are examining whether superlinear convergence is always preferable to linear convergence.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are attempting to clarify the definitions of linear and superlinear convergence, questioning the relationship between the two types of convergence. Some are exploring whether linear convergence implies superlinear convergence and discussing examples of sequences that fit different convergence criteria.

Discussion Status

The discussion is active, with participants providing insights and prompting further exploration of the definitions and relationships between convergence types. There is no explicit consensus, but several lines of reasoning are being examined.

Contextual Notes

Participants are working with definitions and examples from external sources, indicating a reliance on provided materials to frame their understanding of convergence. There may be assumptions about the sequences being discussed that have not been fully articulated.

haya
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Homework Statement



[PLAIN]http://im2.gulfup.com/2011-04-01/1301686351321.gif

Homework Equations



superlinearly convergence

The Attempt at a Solution



[PLAIN]http://im2.gulfup.com/2011-04-01/1301686616101.gif

this is what i know about it, kindly help me
 
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You should look up the definition of a linearly convergent sequence, and then see if one implies the other. P.ex., if a sequence is linearly convergent, then it is also superlinearly convergent (which would be b in your answers). Or, there are sequences which are linearly convergent, but not superlinearly convergent, vice versa, and some which are both (that would be c). And so on. Of course, only one is right.
 
Ok, so assume you have a superlinearly converging sequence {p_n}. Then {c_n} is a zero sequence, so it converges to zero. So, especially (from some n on), it must be smaller than a constant M, right? What does that imply for your original sequence {p_n} in terms of linear convergence?
 

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