Computing Upper & Lower Limit of {Sn} from Expression

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Homework Help Overview

The discussion revolves around computing the upper and lower limits of the sequence {S_n}, defined recursively with specific rules for even and odd terms. The original poster seeks to derive these limits directly from the sequence's definition rather than from the computed terms.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss separating the even and odd terms to simplify the analysis of the sequence. There is an exploration of rewriting the recursive definitions to facilitate this separation.

Discussion Status

Some guidance has been offered regarding the separation of terms, with hints provided to help the original poster structure their approach. The conversation indicates a productive direction, though no consensus or resolution has been reached.

Contextual Notes

The original poster expresses a desire to avoid deducing limits from computed terms, indicating a preference for a more direct analytical approach. There is an implication of ongoing exploration of the recursive definitions.

Ka Yan
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How can I compute the Upper and Lower limit of {Sn}, which defineded as: S1 = 0, S2m = S2m-1 /2, S2m+1 = 1/2 + S2m , directly from its expression, rather than by deduction of the terms?

(i.e., from the definition of Sn, instead of from 0, 0, 1/2, 1/4, 3/4, ...)

thks!
 
Last edited:
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Hi, Ka Yan! :smile:

Hint: deal with the even and odd terms separately.

That is, get an equation with only even terms in it, and another with only odd terms in it.

Instead of one series with two rules, that should give you two series each with only one rule, which is much easier! :smile:
 
I got you!
That you mean, write S2m+1 = 1/2 + S2m-1 /2, and S2m+2 = (1/2 + S2m) /2 instead, isn't it?

Thank you!
 
Last edited:
You're very welcome! :smile:

I assume you can finish it now, but if you can't after a few hours, come back for another hint! :smile:

[size=-2]ooh, you've worked out how to use "size"! :smile:[/size]​
 

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