Stolz Theorem

To find limits of indeterminate expressions xnyn\frac{{{x}_{{n}}}}{{{y}_{{n}}}} of type \frac{{\infty}}{{\infty}} often can be useful following theorem.

Stolz Theorem. Suppose that sequence yn+{y}_{{n}}\to+\infty and starting from some number with increasing of n{n} also increases yn{y}_{{n}} (in other words if m>n{m}>{n} then ym>yn{y}_{{m}}>{y}_{{n}}). Then limxnyn=limxnxn1ynyn1\lim\frac{{{x}_{{n}}}}{{{y}_{{n}}}}=\lim\frac{{{x}_{{n}}-{x}_{{{n}-{1}}}}}{{{y}_{{n}}-{y}_{{{n}-{1}}}}} if limit of the expression on the right side exists (finite or infinite).

Example 1. Find limann\lim\frac{{{{a}}^{{n}}}}{{n}} where a>1{a}>{1}.

Let xn=an{x}_{{n}}={{a}}^{{n}} and yn=n{y}_{{n}}={n} then yn{y}_{{n}} is increasing and yn{y}_{{n}}\to\infty therefore we can apply Stolz Theorem.

limann=limxnyn=limxnxn1ynyn1=limanan1n(n1)=lim(anan1)=\lim\frac{{{{a}}^{{n}}}}{{n}}=\lim\frac{{{x}_{{n}}}}{{{y}_{{n}}}}=\lim\frac{{{x}_{{n}}-{x}_{{{n}-{1}}}}}{{{y}_{{n}}-{y}_{{{n}-{1}}}}}=\lim\frac{{{{a}}^{{n}}-{{a}}^{{{n}-{1}}}}}{{{n}-{\left({n}-{1}\right)}}}=\lim{\left({{a}}^{{n}}-{{a}}^{{{n}-{1}}}\right)}=

=liman(11a)=+=\lim{{a}}^{{n}}{\left({1}-\frac{{1}}{{a}}\right)}=+\infty.

Example 2. If liman=A\lim{a}_{{n}}={A} find lima1+a2+..+ann\lim\frac{{{a}_{{1}}+{a}_{{2}}+..+{a}_{{n}}}}{{n}} (arithmetic mean of first n{n} values of sequence an{a}_{{n}}).

Let xn=a1+a2++an{x}_{{n}}={a}_{{1}}+{a}_{{2}}+\ldots+{a}_{{n}} and yn=n{y}_{{n}}={n} then yn{y}_{{n}} is increasing and yn{y}_{{n}}\to\infty therefore we can apply Stolz Theorem.

lima1+a2++ann=limxnyn=limxnxn1ynyn1=\lim\frac{{{a}_{{1}}+{a}_{{2}}+\ldots+{a}_{{n}}}}{{n}}=\lim\frac{{{x}_{{n}}}}{{{y}_{{n}}}}=\lim\frac{{{x}_{{n}}-{x}_{{{n}-{1}}}}}{{{y}_{{n}}-{y}_{{{n}-{1}}}}}=

=lim(a1+a2++an)(a1+a2+.+an1)n(n1)=liman1=liman=A=\lim\frac{{{\left({a}_{{1}}+{a}_{{2}}+\ldots+{a}_{{n}}\right)}-{\left({a}_{{1}}+{a}_{{2}}+\ldots.+{a}_{{{n}-{1}}}\right)}}}{{{n}-{\left({n}-{1}\right)}}}=\lim\frac{{{a}_{{n}}}}{{1}}=\lim{a}_{{n}}={A}.

For example, since nn1{\sqrt[{{n}}]{{{n}}}}\to{1} then 1+2+33++nnn1\frac{{{1}+\sqrt{{{2}}}+{\sqrt[{{3}}]{{{3}}}}+\ldots+{\sqrt[{{n}}]{{{n}}}}}}{{n}}\to{1}.