Background
Today in my macroeconomics class my teacher taught us three concepts.
The first is very simple: consumption $c$ is a linear function of national income $y$. Mathematically, $$c = My + b$$
We will call $M$ the marginal propensity to consume, and refer to it accordingly as MPC. MPC can also be thought of as the fraction of income that is spent rather than saved. So MPC is going to be some number less or equal to one but greater than or equal to zero ($0 \leq M \leq 1$).
The second concept we learned is also quite simple. As income $y$ increases, MPC decreases.
The third concept is a little bit more complicated. Let Person 1 has $5$ dollars. Let MPC be universally constant and equal to $\frac{3}{4}$. Now, since MPC $= \frac{3}{4}$, Person 1 spends $\frac{3}{4}$ of his or her income on Person 2. So Person 2 has an income of $\frac{3}{4} * 5 = 3.75$ dollars. Person 2 will now spend $\frac{3}{4}$ of his or her income on Person 3. And so on to infinity. I instantly recognized this as the sum of a geometric series: $$5*\sum_{n=0}^\infty\left(\frac{3}{4}\right)^{n} = 20$$ This makes sense, but only when disregarding the second concept (MPC decreases as $y$ increases). When I asked about it my teacher said that since calculus isn't a prerequisite for the class we won't go any further in depth; we'll just let MPC be a constant to simplify things in an introductory-level macroeconomics class. So I thought about the problem for a while today and realized that if we did account for MPC decreasing as y increases, or in this case, MPC increasing as $n$ increases, then we could write the total money spent as an infinite series of kind of infinite product (??). My model can be found below.
The Main Question
I just finished Calculus III last fall so I've never formally learned anything about infinite products. I'm not even sure if these are infinite products since many of them are finite. Anyways, can the following infinite series that I came up with be solved for convergence? If so, how can I figure out the number it converges too? Long explanations will be appreciated. Keep in mind I have a very coarse knowledge of mathematics beyond Calculus III.
$$\sum_{i=0}^{\infty} \left(\prod_{n=0}^{i} \frac{n + 2}{\sqrt{n^2 + 5n + 7}}\right)$$
Perhaps it would be better if I knew how to solve the infinite product first.
Also, I came up with the function $\frac{n + 2}{\sqrt{n^2 + 5n + 7}}$ as a model of MPC because I figured MPC should start around $\frac{3}{4}$ and increase asymptotically towards $1$ as $n$ approaches $\infty$.
Does any of this make economic sense or am I missing a major concept?