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If I know that $a_{n}=f(n)$, is there any reliable means of finding a non-recursive equation for the partial sum from 1 to some n?

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    In general, no. Do you have a specific series in mind?2012-02-29
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    The answer is no since your function is $C^\infty$ howerever if you function is a Polinomial the series will be divergent.2012-02-29
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    There is no general method. There are methods for certain restriced classes of functions $f(n)$, such as polynomials.2012-03-01

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