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.