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Math Forum » Discussions » sci.math.* » sci.math.independent

Topic: Analysis with series 1/2^2+1/3^2
Replies: 8   Last Post: Aug 9, 2012 12:17 PM

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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: Analysis with series 1/2^2+1/3^2
Posted: Aug 9, 2012 12:17 PM
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On Thu, 09 Aug 2012 13:47:18 +0100, Frederick Williams
<freddywilliams@btinternet.com> wrote:

>pholman50@gmail.com wrote:
>
>> Any series can be fluffed out to make it more challenging.
>> Unfluffed, this one is 1/(n^2+n)........1/2 + 1/6 + 1/12 + 1/20......
>> = 1/2 + (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) ......
>> = 1
>> Phil H
>
>May one rearrange 1/2 + (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) ... to
>get (1/2 + 1/2) + (- 1/3 + 1/3) + (- 1/4 + 1/4) + - 1/5 ...?

Yes. Simply look at the partial sums. The sum of the first n terms
on the left side _is_ 1 - 1/n (or 1 - 1/(n+1) or something),
which tends to 1 as n tends to infinity.

But it's good to ask about this, because of the standard example

1 = 1 + (-1+1) + (-1+1) + ...
= (1-1) + (1-1) + ...
= 0.

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