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Infinite series to calculate integrals

Problem Statement
I've been doing maths for fun (I've only learned calculus 1 in university yet, but I love experimenting) and I've found the formula for integration by parts can be generalized as an infinite series.
Relevant Equations
I'm not well educated in infinite series, this is an amateur attempt but it works in most cases. Can someone tell me if this is right? I know in most cases it only makes things more complicated. I've tried it with simpler functions and it seems to work, more complicated ones are sometimes a problem. Here d^0/dx^0 f(x)=f(x)
For example integral of f(x)=sqrt(1-x^2) from 0 to 1 is a problem, since the derivative of the function is -x/sqrt(1-x^2) so putting in 1 in the place of x ruins the whole thing.
 

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The study of power series (Taylor series) is central to the subject of complex analysis. The issues of convergence, derivatives, and integrals are handled in detail there. You might be interested in that subject.
Your equation seems to have some problems. For one, the variable x inside the integral on the left is a dummy variable, but not on the right. Also, consider the series for the exponential function. The integral is the exponential function again and it does not have the alternating sign that your equation has.
 
Thank you for your answer! It's good to read the opinion of someone who is more educated than myself. For now I'm just an enthusiast who enjoys experimenting with maths :)
 
I've found if you plug in e^x the series gives you e^x-1, and by definition integral of e^x is e^x+C, so the derivative of e^x-1 is still e^x. But the series is not great for calculating definite integral, the results will be false. It works for some functions though.
 

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