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is there any definite method or algorithm,software to get exact function or expression from series.e.g we get series solution of differential equation and we want exact expression rather than approximation series ?

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    $\begingroup$ Usually I do this the other way around. I have a power series, and want its sum, so if I can find a DE it satisfies, then solve the DE to get the result! $\endgroup$
    – Gerald Edgar
    Commented Sep 11, 2011 at 12:50

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In some cases yes, see:

http://www.reduce-algebra.com/docs/qsum.pdf

and references therein...

EDIT A canonical reference is Petkovsek-Zeiberger's A=B.

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  • $\begingroup$ More stuff in Wilf's book generatingfunctionology $\endgroup$
    – Gerald Edgar
    Commented Sep 11, 2011 at 12:49

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