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Suppose that we have a power series: $$f(x) = \sum_{k=0}^\infty{c_i x^i}$$

Now suppose that we can get a closed form, e.g. a function composed of simple arithmetic of elementary functions, for the first $N+1$ terms of the series: $$g(x) = \sum_{k=0}^N{c_i x_i}$$

What are the known applications of this? More specifically, I'm looking for the most important/noteworthy application(s). I hope that this avoids being downvoted, as I'm really just trying to get a good intuition of "truncated" series. Any help on asking a better question would be appreciated, because I really would like to get to know this topic better.

Some other questions that asked for an "implementation" or specific series are Constructing finite versions of arbitrary series and Closed form of a partial sum of the power series of exp(x). I'm not looking for how to do it, just what the results could possibly do.

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"...e.g. a function composed of simple arithmetic of elementary functions..." - the thing is, the explicit expressions for such truncations are often not elementary. But they are still results, nevertheless. –  J. M. Jul 10 '12 at 6:00
    
I'd like to know about any truncated series. I am supposing that we can get an "elementary" closed form, whether we actually can or not. What would be the possible consequences? If it's proved that we can't, we can ignore that possibility. –  Matt Groff Jul 10 '12 at 6:07
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Usually what you want to do with such a truncated series is use it as an approximation for the infinite sum $f(x)$. This may be nicer in some way if you have a closed form. In particular, it may be easier to bound the truncation error $f(x) - g(x)$. Or perhaps it may be easier to check whether the error is positive or negative.

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