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May 19th, 2013, 01:06 AM   #1
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Uniform convergence of a serie

My earlier post was about to prove the series is convergent if and divergent if .

So now I have got this problem to show the series is uniform convergent as if .

If I am trying ti use the Weierstrass Test I get annoyed when I could not continue what to do the next step when it was wrong. Could you please show me what to do? What I did was, .

Since is not convergent, the abovementioned series couldn't be uniform convergent. Which is why I did something wrong.
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May 19th, 2013, 08:23 AM   #2
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Re: Uniform convergence of a serie

Quote:
Originally Posted by Noworry
My earlier post was about to prove the series is convergent if and divergent if .

So now I have got this problem to show the series is uniform convergent as if .

If I am trying ti use the Weierstrass Test I get annoyed when I could not continue what to do the next step when it was wrong. Could you please show me what to do? What I did was, .

Since is not convergent, the abovementioned series couldn't be uniform convergent. Which is why I did something wrong.
We have already prove that is convergent for

To prove that is uniformly convergent let us choose some fixed value such that so we have the following

.

Hence the series is uniformly convergent on
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May 20th, 2013, 06:46 AM   #3
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Re: Uniform convergence of a serie

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Originally Posted by zaidalyafey
We have already prove that is convergent for

To prove that is uniformly convergent let us choose some fixed value such that so we have the following

.

Hence the series is uniformly convergent on
We know that the earlier series converges as . So we call another value r that contains in the same interval, that is . If we look away from the enequality, so this series looks similar to the first series if we let r = p. Unfortunately the interval of its convergence would exactly be same, so it would be wrong. BUT if we look at this enequality, the convergence of interval as you mentioned would be true. Could you please elaborate a bit more because I still feel like I don't understand it completely {I still doubt}?
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May 20th, 2013, 07:33 AM   #4
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Re: Uniform convergence of a serie

Hopefully an example will clear the doubt

Prove that is uniformly convergent on

since we know that is convergent forall , it must be convergent for
, right ?

So let us now apply the M-test



Hence is uniformly convergent on

Now , can we generalize that for
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