I was exploring the euler-mascheroni constant when I thought of this problem. The euler-mascheroni constant (usually denoted as $\gamma$) can be calculated a number of ways, but the primary example is this formula: $$\lim_{x\to\infty}\ln(x)-\sum_{n=1}^{x}\frac{1}{n}=-\gamma$$ I thought about what other kinds of constants I can generate. I remembered that $\frac{d}{dx}\ln(x)=\frac{1}{x}$, so I first thought to generalize the "constant generating method" like this: $$\lim_{x\to\infty}f(x)-\sum_{n=1}^{x}f'(n)$$ where $f(x)$ and its derivative can be evaluated for all positive integers, and the limit is discrete (meaning $x$ is only a positive integer when it approaches infinity). For the rest of the question, we'll call it The Operation${}^{{}^{\mathsf{TM}}}$. Of course, this diverges for (I estimate) most functions. For example, when $f(x)=x^2$, the value of the above operation tends to negative infinity. But for other functions that grow more slowly or tend to zero, it can converge. For example, when $f(x)=\ln(x)$, you get the negative of the euler-mascheroni constant (by defenition), and when $f(x)=\frac{1}{x}$, you get essentially the Basel problem which evaluates to $\frac{\pi^2}{6}$. One that I find quite interesting is when $f(x)=\operatorname{W}(x)$ where $\operatorname{W}(x)$ is the Lambert-W funciton (AKA the product-log function), it evaluates to about 0.3688... when I added the first 5000 terms into Wolfram-Alpha. The point to take away from this is that for some functions, the operation will converge to a constant.
So after toying around this a bit, I came to my question: What condition(s) are needed for $f(x)$ so that the operation converges? Assume the limit is always discrete.
Some conditions I am looking for (proof required):
- Examples of specific functions given a parameter (e.x. the operation converges for the form $f(x)=x^a$ for $a\leq1$)
- Proof of convergence or divergence given a property (e.x. if $\lim_{x\to\infty} f'(x)=0$, then the operation converges)
- Proof of a property given convergence or divergence (e.x. if the operation converges, then $\lim_{x\to\infty} f'(x)=0$)
- A comparison test (e.x. $\ln(x)$ converges under the operation. $\lim_{x\to\infty}\operatorname{W}(x)<\ln(x)$, therefore $\operatorname{W}(x)$ also converges)
- If at all possible, a complete condition (e.x. the operation converges if and only if $\lim_{x\to\infty} f'(x)=0$)
I haven't been able to prove any of the examples. If this is something known, please share it with me! And if any of you can provide research on this, please share it with me in the comments or an answer. I am very interested in this because I haven't found it anywhere before. The most important (for lack of a better term) or interesting property receives the answer reward.
*post-question note: I find it kind of interesting how the operation takes in a whole function as an input, and a number as an output. It's like a function-function.
