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What are the known ways that allow us in some cases to precisely calculate the real roots of some specific polynomials by using infinite sums? I'd appreciate any reference on this topic.

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    The roots of a polynomial of degree five and higher can be expressed in terms of either theta functions or hypergeometric functions; these might be the "infinite sums" you had in mind. Look up the Bring radical, for instance.2012-06-09
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    Depends on what you mean by "infinite sums." Do the methods described at http://en.wikipedia.org/wiki/Root-finding_algorithm count?2012-06-09
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    @Qiaochu Yuan: thanks for reference! I'll check it soon.2012-06-09

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