Finding Upper and Lower Limits of Sn

[Ka Yan]

How can I compute the Upper and Lower limit of {Sn}, which defineded as: S1 = 0, S2m = S2m-1 /2, S2m+1 = 1/2 + Sm , 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!

(I'm sorry, erm, I post this question here. I had moved it into "Precalculus Mathematics" of " Homework & Coursework Questions" .

Sorry, manager.)

 

[Mark44]

How can I compute the Upper and Lower limit of {Sn}, which defineded as: S1 = 0, S2m = S2m-1 /2, S2m+1 = 1/2 + Sm , 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!

Better late than never ...
Look at each subsequence separately, since there are different formulas for the odd-subscript terms and the even-subscript terms.
For example, for the odd terms,
$s_1 = 0$
$s_3 = \frac 1 2 + s_1$
$s_5 = \frac 1 2 + s_2 = \frac 1 2 + \frac 1 2 + s_0$
Continue the process until you see a pattern. Try a similar technique for the even-subscript terms.