I am struggling to find an answer of the following series
$$\sum_{i=1}^n \frac{1}{1+\exp(a_i+b_ix)}$$
Any suggestion?
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Any information what $a_i$ and $b_i$ are? – Fabian Oct 05 '12 at 15:20
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This is no series as it has only $n$ terms... What exactly do you want to calculate? – Sh4pe Oct 05 '12 at 15:25
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I'm confused by the question. What makes you think this series converges? What's the motivation here? It will help to know that. – 000 Oct 05 '12 at 15:27
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I don't think you will find an expression in terms of $\exp(\sum a_i)$ and $\exp(\sum b_i)$ Note that if one $a_i$ is very large, the corresponding entry into the sum is essentially zero and doesn't matter. When you add all the $a_i$ it will dominate. – Ross Millikan Oct 05 '12 at 22:48
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$a_i$ and $b_i$ are any number in R. I need to find an equal expression which includes $e^{\sum _i^n a_i}$ and $ e^{\sum _i^n b_i}$ – Oct 05 '12 at 22:38
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It should be $\frac{1}{1+exp(\sum a_i+\sum b_i x)}*f(\sum a_i,\sum b_i x)$ . problem is finding function $f$ – Oct 05 '12 at 23:25
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You won't be able to. For an example, let all the $b_i$ be zero. Compare $a_1=a_2=5$ to $a_1=0, a_2=10$ These are not equal, but your expression cannot tell them apart. – Ross Millikan Oct 05 '12 at 23:36