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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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    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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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.