Fascinating Results of Infinite Integration by Parts

Question

This is NOT the same topic as the post "Infinite Integration by Parts", though it is similar.

I've noticed some truly fascinating results of expressing infinite integration by parts as an infinite series. For example,

$$\int \sin(x)\,dx = C +\sum_{n=1}^\infty{(-1)^{n-1}\frac{x^{2n-1}\sin(x)}{(2n-1)!}+(-1)^n\frac{x^{2n}\cos(x)}{(2n)!}}$$

Simply decrementing the index by one gets you

$$C+\sum_{n=0}^\infty{(-1)^n\frac{x^{2n+1}\sin(x)}{(2n+1)!}+(-1)^{n+1}\frac{x^{2n}\cos(x)}{(2n)!}}=-\cos(x)+C+\sum_{n=1}^\infty{(-1)^{n-1}\frac{x^{2n-1}\sin(x)}{(2n-1)!}+(-1)^n\frac{x^{2n}\cos(x)}{(2n)!}}$$ The sum is convergent for any finite value of $x$, and the antiderivative of $\sin(x)$ is indeed $-\cos(x)+C$. I tried the same process on a few other functions that have elementary antiderivatives, with similar results.

Here's where things get really interesting. For $\frac{1}{\log(x)}$, whose antiderivative is not elementary, I got $$\frac{x}{\log(x)}+C+\sum_{n=1}^\infty{\frac{(n-1)!x}{\log^{n}(x)}}$$ and for $x$, which is not infinitely differentiable, I got $$C+x^2 \sum_{n=0}^\infty{(-1)^n}$$ by this same process of infinite integration by parts and decrementing the initial index of the resulting series.

Various summation methods for divergent series, such as Cesaro summation, assign $\sum_{n=0}^\infty{(-1)^n}$ a value of $\frac{1}{2}$, and $\frac{1}{2}x^2$ is the antiderivative of $x$. Notice also that the zeroth term of the series for $\frac{1}{\log(x)}$ is $\frac{x}{\log(x)}$, and that $\frac{x}{\log(x)}$ ~ $li(x)+C = \int \frac{dx}{\log(x)}$.

It appears that this process yields convergent series for functions with elementary antiderivatives and divergent series for those without them (I tried it on other functions with non-elementary antiderivatives as well, such as $x^x$.) Maybe a divergent summation method for $\sum_{n=1}^\infty{\frac{(n-1)!x}{\log^{n}(x)}}$ could yield a function that equals $li(x) - \frac{x}{\log(x)}$, which I'm sure would be of intense interest to number theorists, if it hasn't already been found. Has any of this been investigated by anyone else? Is it actually of any use, or am I seeing patterns where they don't exist?

Answer 1

The hypothesis that repeated integration by parts yields a divergent series for non-elementary functions is false, since it yields a convergent one for $e^{x^2}$. It would perhaps be more fruitful to try repeated integration by parts on $\sin^{m}{x}$ since it yields some interesting results (Wallis formula etc).