Given that the Minkowski distance between points $X = (x_1,\dots,x_n)$ and $Y = (y_1,\dots ,y_n)$ is defined as following
$$
d(X, Y)
= \left(\sum_{i=1}^n|x_i−y_i|^p\right)^{1/p},
$$
I learned that as p goes to the infinity our result will be max distance between the points. Defined as following ;
$$
\lim_{p\to\infty}d(X, Y) = \max_{i=1, \dots, n} |x_i-y_i|.
$$
I want to learn that what will happen as n (the number of point)goes to the infinity? What will be the results of this limit ?
By the way p can be any integer value starting from 1 to infinity. But to make it simple, lets take p as 2.
My question is as follows;
$$ \lim_{n\to\infty} \left(\sum_{i=1}^n|x_i−y_i|^p\right)^{1/p}=? $$
The answer is easy:
$$\lim_{n\to\infty} \left(\sum_{i=1}^n|a_i|^p\right)^{1/p}=\left(\sum_{i=1}^{\infty}|a_i|^p\right)^{1/p}, $$
if $(a_i) \in l^p$.