Computing Upper & Lower Limit of {Sn} from Expression

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To compute the upper and lower limits of the sequence {Sn}, defined recursively, it is suggested to separate the even and odd terms into distinct equations. This approach simplifies the analysis by allowing the formulation of two series, each governed by a single rule. Specifically, the equations can be expressed as S[SIZE="1"]2m+1 = 1/2 + S[SIZE="1"]2m-1 /2 for odd terms and S[SIZE="1"]2m+2 = (1/2 + S[SIZE="1"]2m) /2 for even terms. This method facilitates a clearer understanding of the sequence's behavior without relying on the initial term deductions. Following this guidance should enable a successful computation of the limits.
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!
 
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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!
 
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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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