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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? –  Aryabhata Feb 29 '12 at 23:37
    
The answer is no since your function is $C^\infty$ howerever if you function is a Polinomial the series will be divergent. –  checkmath Feb 29 '12 at 23:39
    
There is no general method. There are methods for certain restriced classes of functions $f(n)$, such as polynomials. –  André Nicolas Mar 1 '12 at 0:03

1 Answer 1

See e.g. "Gosper's algorithm" in the case of hypergeometric terms for which the sum can be expressed as $S(n) - S(0)$ where $S(n)$ is a hypergeometric term.

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