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Topic: Power Series (Read 652 times) |
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Braincramps
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Power Series
« on: May 16th, 2004, 2:04am » |
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Suppose f(r)= [smiley=csigma.gif][smiley=infty.gif] n=1anrn has (-1,1) as its interval of convergence (f may or may not converge at endpoints) and f(r) has a finite limit as r [smiley=longrightarrow.gif]1-. Let N(r) be a positive integer-valued function such that N(r)[smiley=longrightarrow.gif][smiley=infty.gif] as r[smiley=longrightarrow.gif]1-. Does [smiley=csigma.gif][smiley=infty.gif]n=N(r) anrn have a finite limit as r[smiley=longrightarrow.gif]1-?
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Barukh
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Re: Power Series
« Reply #1 on: May 16th, 2004, 9:17am » |
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I just wanted to clarify:
Is it the case, that when n runs through all the possible values of N(r), r also runs through the whole convergence interval, so that the last series is no longer a power series?
Also, it seems that N(r) will take the same value more than once (in fact, infinitely often) - what terms of the series are taken then?
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TenaliRaman
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I am no special. I am only passionately curious.
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Re: Power Series
« Reply #2 on: May 16th, 2004, 9:34am » |
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My initial thoughts :
sum(n=N(r) to inf)anrn
= sum(n=0 to inf)anrn - sum(n=0 to N(r))anrn ... (*)
Now i know that convergence is not affected in a power series as long as we remove finite number of terms from the start of the power series. So as long as N(r) is finite , the * converges.
now that i read Barukhs post, i am forced to make some more thinking over it , whether or not taking limit r->1- on (*) would be proper since N(r) isn't continuous ( )
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Self discovery comes when a man measures himself against an obstacle - Antoine de Saint Exupery
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Braincramps
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Re: Power Series
« Reply #3 on: May 16th, 2004, 10:28am » |
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on May 16th, 2004, 9:17am, Barukh wrote:| Is it the case, that when n runs through all the possible values of N(r), r also runs through the whole convergence interval, so that the last series is no longer a power series? |
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r doesn`t vary as n varies - but the number of terms of the series vary as r varies. As an example we could take N(r), the lower limit of the sum at r, to be an increasing step function satisfying 1/ [smiley=surd.gif](1-r) < N(r) < 1/(1-r).
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| « Last Edit: May 16th, 2004, 10:38am by Braincramps » |
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