My friends and myself were discussing Pascal's Triangle, specifically the following property of it.
First, consider the Pascal's Triangle - $$1\\ 1\ 1\\ 1\ 2\ 1\\ 1\ 3\ 3\ 1\\ 1\ 4\ 6\ 4\ 1\\ 1\ 5\ 10\ 10\ 5\ 1\\ 1\ 6\ 15\ 20\ 15\ 6\ 1\\ ..................\\ .....................$$
Now, one interesting observation that can be easily spotted is that initial rows of the Pascal's Triangle follow the form $11^n$ and then a few of the following rows follow $101^n$ and so on.
Now, if we deform the above argument as the following:
- The zeroth row is of form $(1.1)^0$
- The first row is of the form $(1.1)^1$
- Similarly the second row as $(1.1)^2$ the following one as $(1.1)^3$ and then the fourth row as $(1.1)^4$
- Now for the immediate next row if we follow the pattern shown above we get overflow due to digits being carried forward and hence we represent the fifth row as $(1.01)^5$ yielding $1.0510100501$.
- Now again we keep on proceeding with the introduction of more zeroes after the decimal point for preventing overflow and preserving the form of the Pascal's Triangle.
Consider the following idea, the $n^\text{th}$ row can be represented as following $(1.\overbrace{000.....0}^{n}1)^n$. Now I grossly miscalculated this, previously. As $n \to \infty$ this value becomes equal to $1$.
Due to the fatal error above, as pointed out in the answers, I rephrase the initial question as: Is there any way to make this series converge to e with the adding of zeroes suitably as needed?
Also, later on studying the Pascal's Triangle further I found another interesting relation.
Consider, $f(n)$ to be the product of all digits in the $n^\text{th}$ row of the Pascal's Triangle. Also, consider the beginning to be the zeroth row. Then, with algebraic manipulation we obtain: $$\frac{f(n)}{f(n-1)} = \frac{n^{(n-1)}}{(n-1)!}$$ from which we can further deduce that $\frac{f(n+1) \times f(n-1)}{f(n)^2}$ converges to $e$ as $n \to \infty$ i.e., $$\lim_{n \to \infty} \frac{f(n+1) \times f(n-1)}{f(n)^2} = e$$
Since, I have updated the problem statement, I am not very sure if the following questions hold .
- can we connect the initial observation with above observation?
- is there some correlation between the number $e$ and the product of its digits? (Well this seems quite silly now :))
Update: Thanks for pointing out the mistake.