Understanding Series Convergence and Changing n=1 to n=1+k

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Homework Help Overview

The discussion revolves around the convergence of series and the implications of changing the index of summation from n=1 to n=1+k. Participants are exploring whether such a change affects the limit of convergence.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning the effects of modifying the index of summation on series convergence and whether there are general rules governing this change.

Discussion Status

Some participants have provided insights into the nature of convergence and the treatment of dummy variables in series. There is an acknowledgment of intuitive ideas regarding the topic, but no explicit consensus has been reached.

Contextual Notes

One participant references the need for rigorous statements found in analysis literature, suggesting that formal definitions and rules may be under discussion.

ptolema
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this is just a general question: do
conv.jpg
both converge to the same limit? is there a general rule for changing n=1 to n=1+k?
 
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Convergence of a series is not affected if you add/remove finitely many terms to it. A more rigorous statement can be found in any of the many elementary books on analysis, for e.g. W. Rudin's Principles of Mathematical Analysis.
 
alright, thanks! i had an intuitive idea, but i wasn't sure if there was something i had to do to an first. makes a lot more sense now
 
Yes. The other way would be to replace n by m, where m = n+k (so that the range of summation changes from 0 to Inf -> k + Inf); Since m is a dummy variable, we can as well call it n.
 

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