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Is there a general way to find the general form of a series?

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There are heuristics; certain things (like alternating signs, only even terms are nonzero, etc) show up a lot and there are "standard tricks" to achieve them. But for an arbitrary series, it may be very difficult to do. In a sense, it is impossible to have an algorithm/recipe to do it, because there are far more series than there are formulas we can write down for them. –  Arturo Magidin Oct 8 '11 at 21:35
Could you please explain a little bit more about what you mean by standard tricks? –  geraldgreen Oct 8 '11 at 21:36
What do you mean by "find the general form of a series"? Maybe you can illustrate with an example of a series together with its "general form". –  Gerry Myerson Oct 8 '11 at 21:45
Alternating signs can be achieved with a factor of $(-1)^n$ or $(-1)^{n+1}$ (depending on whether you want terms with even $n$ positive, or terms with even $n$ negative, respectively). If you only want even-indexed terms, you usually index by $2n$ instead of by $n$; if you only want odd-indexed terms, then you index with $(2n-1)$ or with $(2n+1)$. That's the kind of thing I'm refering to. But these are things that one picks up with experience and practice, not a recipe that you can be given. –  Arturo Magidin Oct 8 '11 at 21:48
Do you mean that you want a mechanical approach to find definite or indefinite sums, à la Gosper's algorithm and variants (q-analogues, etc)? –  Peter Taylor Oct 8 '11 at 21:58

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