# Thread: Problem with the expansion of integration by parts

1. I've come across this funny problem while messing around with integration by parts. Probably made a mistake somewhere.

In the integration of parts expression, it's possible to expand it further.

Plugging the second expression into the first, we get

I don't think this is standard notation, but it's better than writing out all the "violin holes".

The problem is that it seems like we will get an infinite string of polynomials on the right hand side, but not on the left.

2. What is the point, exactly ?

3. Originally Posted by Markus Hanke
What is the point, exactly ?
it seems like we will get an infinite string of polynomials on the right hand side, but not on the left.

4. Originally Posted by iopst
it seems like we will get an infinite string of polynomials on the right hand side, but not on the left.
Yes, but that's perfectly fine so long as they are equal. Have you tried to check whether the sum converges, i.e. whether it has a well defined limit ?

5. -

6. Originally Posted by Markus Hanke
Yes, but that's perfectly fine so long as they are equal. Have you tried to check whether the sum converges, i.e. whether it has a well defined limit ?
sigh, my maths skills are not good enough for me to find out.

7. Originally Posted by iopst
it seems like we will get an infinite string of polynomials on the right hand side, but not on the left.
That mess on the right is not a polynomial.

Markus Hanke's question of convergence is legitimate, but since the functions in question are rather arbitrary, there is no reason to think that the series is likely to converge. In general it will not. In fact since one need not deal with infinitely differentiable functions in applying the technique of integration by parts, the series may very well cease to make sense after some number of terms.

A more important question is why anyone would want to construct such a series. In some cases one does want to integrate by parts a couple of times, but I know of no problem of interest where such an infinite series would produce anything other than a headache.

Integration by parts is really nothing more than an application of the product rule for derivatives along with the fundamental theorem of calculus. It is a useful tool when used appropriately, but this does seem to be such a case.

8. Besides, its not correct to say that in general , v = integral v dx !
Remember we don't know what v is !

If it was the peculiar case where v = integral v dx , then it looks correct.

9. Originally Posted by DrRocket
That mess on the right is not a polynomial.

Markus Hanke's question of convergence is legitimate, but since the functions in question are rather arbitrary, there is no reason to think that the series is likely to converge. In general it will not. In fact since one need not deal with infinitely differentiable functions in applying the technique of integration by parts, the series may very well cease to make sense after some number of terms.

A more important question is why anyone would want to construct such a series. In some cases one does want to integrate by parts a couple of times, but I know of no problem of interest where such an infinite series would produce anything other than a headache.

Integration by parts is really nothing more than an application of the product rule for derivatives along with the fundamental theorem of calculus. It is a useful tool when used appropriately, but this does seem to be such a case.
It will have powers of x, but it will be more much more complicated than an ordinary polynomial, so that's why there's no contradiction.

I have no ends in mind, just playing around with formulas.

10. Originally Posted by Isilder
Besides, its not correct to say that in general , v = integral v dx !
Remember we don't know what v is !

If it was the peculiar case where v = integral v dx , then it looks correct.
Yup. I've managed to express the multiple integrals in terms of my own funny notation.

11. I'm confused how you got there exactly...

It seems like you repeatedly substituted something in, and then rewrote it as a series?

12. To chime in here, the "repeated product rule" trick on nice functions (with a little bit of finesse) generates the Taylor series with the integral form of the remainder.

I'm too lazy to write this out again so here is my answer from thescienceforum:

Ignoring all the technical details we can start with a nice function that satisfies the fundamental theorem of calculus, so

We can do a substitution and rewrite this equation slightly as and then we start iterating the product rule aka integration by parts.

But we notice we are merely generating the Taylor Series of the Function in question and the integral form of the remainder

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