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Let $\xi_1, \xi_2, \cdots, \xi_n$ be indeterminates. Define the following indeterminates: $s_k := \sum\limits_{i=1}^n\xi_i^k, 1\le k <\infty ,$ $\sigma_k := \sum\limits_{1\le i_1

How to show $ \prod\limits_{i=1}^n(1-\xi_it)=1-\sigma_1t+\sigma_2t^2-\cdots+(-1)^n\sigma_nt^n=\exp\left(-\sum\limits_{j=1}^\infty s_j\frac{t^j}{j}\right)?$

Thanks.

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    @t.b.: If it's any consolation, I find the acronyms most of my students use that are related to math just as confusing (I'm also not a native speaker either). And I positively dislike "FOIL" and its many derivations. As far as I can tell, it actually makes students *unable* to expand products unless they happen to be binomial times binomial...2011-11-06

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The power series of the logarithm gives

$\log (1-\xi t) = -\sum_{k=1}^{\infty} \xi^k \frac{t^k}{k}.$

Summing this identity for the different values of $\xi$ and then taking the exponential gives the desired identity.