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mike lowry
science forum beginner
Joined: 20 May 2006
Posts: 7
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 Posted: Sat May 20, 2006 5:39 am Post subject:
real analysis
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Consider the sequence of functions (fn), where fn(x) = (1/n)sin(x/n)
a) show that the sum of the series fn(x) from n=1 to infinity converges uniformly on [-r,r] for every r > 0.
b) prove or disprove that when integrated term by term, the sum of the series from n=1 to infinity fn(s)ds is pointwise convergent on [-r,r]. |
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Tomas Andersson
science forum beginner
Joined: 06 Jun 2005
Posts: 43
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| Posted: Sat May 20, 2006 3:13 pm Post subject:
Re: real analysis
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On Sat, 20 May 2006 08:39:44 +0300, mike lowry <mmehdiza@gmu.edu> wrote:
| Quote: | Consider the sequence of functions (fn), where fn(x) = (1/n)sin(x/n)
a) show that the sum of the series fn(x) from n=1 to infinity converges
uniformly on [-r,r] for every r > 0.
b) prove or disprove that when integrated term by term, the sum of the
series from n=1 to infinity fn(s)ds is pointwise convergent on [-r,r].
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Don't just post questions and expect us to answer them for you. First show
us what you've tried yourself and why it didn't work. Then we'll help you
if you ask nicely.
/T |
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mike lowry
science forum beginner
Joined: 20 May 2006
Posts: 7
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| Posted: Mon May 22, 2006 12:03 am Post subject:
Re: real analysis
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My problem is Im not sure what theorem to use, whether Abel, or Weisteress M test, or something else.
I couldnt use the Weiss M test because 1/n diverges if I want to keep the series |sin (x/n)|< or = 1. I assume I have to take partial sums, but Im not sure if its with the original function of the derivatives.
Could you please help me. Thank you for your help in advance. |
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richard.blankman@gmail.co
science forum beginner
Joined: 11 Jan 2006
Posts: 44
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| Posted: Tue May 23, 2006 4:49 am Post subject:
Re: real analysis
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mike lowry wrote:
| Quote: | My problem is Im not sure what theorem to use, whether Abel, or Weisteress M test, or something else.
I couldnt use the Weiss M test because 1/n diverges if I want to keep the series |sin (x/n)|< or = 1. I assume I have to take partial sums, but Im not sure if its with the original function of the derivatives.
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You're right, the sum of 1/n as n=1 to infinity diverges. Try using the
limit comparison test with 1/n^2 however....
| Quote: |
Could you please help me. Thank you for your help in advance.
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(For the second problem, just integrate it. That should give you an
answer right away.) |
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cody.roux@gmail.com
science forum beginner
Joined: 30 Apr 2006
Posts: 34
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| Posted: Tue May 23, 2006 1:02 pm Post subject:
Re: real analysis
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mike lowry a écrit :
| Quote: | Consider the sequence of functions (fn), where fn(x) = (1/n)sin(x/n)
a) show that the sum of the series fn(x) from n=1 to infinity converges uniformly on [-r,r] for every r > 0.
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Try approximating the sum by an integral, i.e. using y=1/n, show that
the integral of y*sin(xy)dy with y going from 1 to infinity converges.
You can then use the value of the bounding integral to show the uniform
convergance.
good luck! |
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