Stolz Theorem

To find limits of indeterminate expressions `(x_n)/(y_n)` of type `(oo)/(oo)` often can be useful following theorem.

Stolz Theorem. Suppose that sequence `y_n->+oo` and starting from some number with increasing of `n` also increases `y_n` (in other words if `m>n` then `y_m>y_n`). Then `lim (x_n)/(y_n)=lim (x_n-x_(n-1))/(y_n-y_(n-1))` if limit of expression on the right side exists (finite or infinite).

Example 1. Find `lim (a^n)/n` where `a>1`.

Let `x_n=a^n` and `y_n=n` then `y_n` is increasing and `y_n->oo` therefore we can apply Stolz Theorem.

`lim (a^n)/n=lim (x_n)/(y_n)=lim (x_n-x_(n-1))/(y_n-y_(n-1))=lim (a^n-a^(n-1))/(n-(n-1))=lim(a^n-a^(n-1))=`

`=lim a^n(1-1/a)=+oo`.

Example 2. If `lim a_n=A` find `lim (a_1+a_2+..+a_n)/n` (arithmetic mean of first `n` values of sequence `a_n`).

Let `x_n=a_1+a_2+...+a_n` and `y_n=n` then `y_n` is increasing and `y_n->oo` therefore we can apply Stolz Theorem.

`lim (a_1+a_2+...+a_n)/n=lim (x_n)/(y_n)=lim (x_n-x_(n-1))/(y_n-y_(n-1))=`

`=lim ((a_1+a_2+...+a_n)-(a_1+a_2+....+a_(n-1)))/(n-(n-1))=lim (a_n)/1=lim a_n=A`.

For example, since `root(n)(n)->1` then `(1+sqrt(2)+root(3)(3)+...+root(n)(n))/n->1`.