Does the green area converge to a known constant when $n\to \infty$?

Question

Let $n$ denote the number of the rectangles in the figure above.

We know that the gray area converges to Euler-Mascheroni constant $(\gamma)$ when $n\to \infty$.

I have three questions about the green area:

$1)$ Does it converge to a known constant when $n\to \infty$?

$2)$ Can we relate it to $\gamma$ when $n\to \infty$?

$3)$ Which is one is larger when $n\to \infty$, the gray area or the green one?

I came come up with this question when I was searching about $\gamma$ and saw the image above and the green area brought my attention.

Thank you.

Answer 1

In your picture, we have: $$ \text{gray}+\text{green} = \lim_{n\to{\infty}}{\sum_{j=1}^{n} \left(\frac{1}{j} - \frac{1}{j+1}\right)} $$ This telescopes, showing $$ \text{gray}+\text{green} = 1 $$ Therefore $$ \text{green} = 1-\text{gray} = 1-\gamma $$ Because the graph $1/x$ is convex, we have $\text{green} < \text{gray}$. In fact, $\gamma \approx 0.57$ so $1-\gamma \approx 0.43$.