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I'm attempting a novel approach to some tough integration problems. I'm using the idea of series expansions to help integrate. In other words, I will attempt to approximate integration by integrating the series expansion of an integrand, rather than direct integration or standard numerical methods.

I believe I can approximate integration of a series very easily, compared to the other methods. However, there's a catch. I will use at least two different series expansions. One for the lower limit of integration, and one for the upper limit. Now, when I attempt to integrate these expansions, the constant of integration comes into play, and it's not obvious what it is. Since I am using at least two different series expansions, the constant of integration may differ for each expansion. So I'm wondering if there is an easy way to get the constants of integration without much more work. Any help, ideas, or suggestions are welcome.

EDIT

A few additional notes... I know ahead of time that the series will converge. I consider that I could integrate in sections, like quadrature, while still using the series to aid in integration. However, I am considering the idea of only using only the endpoints, with two different series. So the constants of integration would be different for each series. If I could somehow find them or find how they differ relative to one another, that would save me the trouble of breaking the integral into sections and using something akin to conventional numerical methods.

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    You are calculating a definite integral. You must use the same constant of integration for the lower limit as for the upper limit. Effectively, you need not worry about the constant of integration, any series that has the right derivative will do. But to repeat, you must use the **same** series in calculating at upper limit, lower limit.2011-08-08
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    I think you have in mind to replace the *integrand* by a series approximation, which is certainly an idea worth pursuing. However I don't think your thinking about it is ripe enough for an answer to be given. The constant of integration really won't matter if you are developing an approximation to a definite integral. If you are approximating an antiderivative, then the consistency of a constant across a boundary where you switch the approximating series can be a valid problem. Basically you would choose the constants so the results agree at the boundary where they switch (continuity).2011-08-08
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    Series expansions are sometimes a very useful thing. However, for most nice functions of any complexity, even getting the first few terms is difficult. And there are issues of convergence. Look for example at $\int_3^{11}\frac{1}{1+x} \,dx$. The usual series for $1/(1+x)$ does not converge for $x$ in our interval.2011-08-08
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    @André: you can always use the Laurent series, which is also pretty usual :)2011-08-08
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    @Mariano Suarez-Alvarez: True, it depends on the meaning of "usual." I was trying to point to difficulties at the OP's likely level in his/her studies.2011-08-08
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    @Alice: Sorry for the slow response. An "easy" variation can be found here: http://math.stackexchange.com/questions/117782011-08-08
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    @André Nicolas: I'm not particularly well studied yet. My main concern is if I can use TWO DIFFERENT SERIES rather than one. Unfortunately, I then have to find the constant of integration, or how the constants differ relative to each other, I believe. Another option would be to imitate numerical methods by splitting the integral into sections, which would mean many more series to attempt to integrate (using this particular method).2011-08-08
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    @Matt Groff: I had an incorrect understanding of the level of the question. The link you sent cleared it up. There is a potentially very painful process of "matching,"2011-08-08
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    @André Nicolas: I'm interested in "matching". I'd like to hear basically any/all potential solutions to the problem. Please feel free to contribute - your help is much welcomed and appreciated!2011-08-08

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If you are talking about looking for the anti-derivative (though you also mentioned upper and lower limits), then you just write some constant $C$ after all your calculations. E.g. if $$ f(x) = \sum\limits_{k=0}^\infty a_k x^k $$ then $$ F(x) = \int f(t)\,dt = \sum\limits_{k=0}^\infty\frac{a_k}{k+1}x^{k+1} +C. $$ The idea of an anti-derivative is the following: $$ \int\limits_{a}^bf(t)\,dt = F(b)-F(a) $$ for any anti-derivative $F$ of the function $f$. Since any anti-derivative is determined up to a constant, you just should pick one anti-derivative before calculating $F(b)-F(a)$. In our example, you can put $C=1$ or $C=2$ and the result will stay the same.

Finally, to apply such a technique you should be careful since you cannot always integrate series in such a simple way - there are sufficient conditions to integrate series by parts.