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manankhurma
 Post subject: Summation of a non-standard series
PostPosted: Wed, 7 Sep 2011 15:20:00 UTC 
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In terms of a_1,a_2,a_3,...,a_n, find the sum of

\frac{a_1}{1+a_1} + \frac{a_2}{(1+a_1)(1+a_2)} + \frac{a_3}{(1+a_1)(1+a_2)(1+a_3)} + ... + \frac{a_n}{(1+a_1)(1+a_2)...(1+a_n)}
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outermeasure
 Post subject: Re: Summation of a non-standard series
PostPosted: Wed, 7 Sep 2011 15:26:31 UTC 
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manankhurma wrote:
In terms of a_1,a_2,a_3,...,a_n, find the sum of

\frac{a_1}{1+a_1} + \frac{a_2}{(1+a_1)(1+a_2)} + \frac{a_3}{(1+a_1)(1+a_2)(1+a_3)} + ... + \frac{a_n}{(1+a_1)(1+a_2)...(1+a_n)}


Please show your work.

A small hint towards a solution: \dfrac{a_1}{1+a_1}=1-\dfrac{1}{1+a_1}.

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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manankhurma
 Post subject: Re: Summation of a non-standard series
PostPosted: Wed, 7 Sep 2011 15:40:32 UTC 
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Hi, "Please show your work"? What work?
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manankhurma
 Post subject: Re: Summation of a non-standard series
PostPosted: Wed, 7 Sep 2011 15:43:12 UTC 
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That helped. Thanks
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Shadow
 Post subject: Re: Summation of a non-standard series
PostPosted: Wed, 7 Sep 2011 16:20:15 UTC 
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manankhurma wrote:
What work?


Exactly. We're asking you to show us what you've done toward solving the problem, so that we're not just parroting information to you that you could have gotten yourself with a little effort.

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Denis
 Post subject: Re: Summation of a non-standard series
PostPosted: Wed, 7 Sep 2011 16:35:42 UTC 
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Not just that, Manan: that could be a problem given to you by your math teacher,
and you're asking us to do your homework :wink:

If you ask: does that happen? Answer is: VERY often :shock:

Stick around and act as tutor for a while...you'll see....

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