This page contains the Stolz's theorem.

This theorem is very useful when one can find a limit of expressions x_{n}/y_{n} that are of ∞/∞ type.

Let **y _{n} → +∞**. Moreover, starting from some point

y_{n+1} > y_{n}

then

lim x_{n}/y_{n} = lim [x_{n} - x_{n-1}]/[y_{n} - y_{n-1}]

if only the limit on the right hand side exists.

Let us assume that the rhs limit equals a finite number **L**. Then for **ε > 0** there is such an index **N** that for **n > N** is

|[x_{n} - x_{n-1}]/[y_{n} - y_{n-1}] - L| < ε/2

and

L - ε/2 < [x_{n} - x_{n-1}]/[y_{n} - y_{n-1}] < L + ε/2.

It means that for any **n > N** all fractions

[x_{N+1} - x_{N}]/[y_{N+1} - y_{N}], [x_{N+2} - x_{N+1}]/[y_{N+2} - y_{N+1}], …
[x_{n-1} - x_{n-2}]/[y_{n-1} - y_{n-2}], [x_{n} - x_{n-1}]/[y_{n} - y_{n-1}]

are between **L - ε/2** and **L + ε/2**. Because from some point

y_{n+1} > y_{n}

among them is also the fraction (obtained as a sum of all nominators and denominators)

[x_{n} - x_{N}]/[y_{n} - y_{N}]

and for **n > N** is

|[x_{n} - x_{N}]/[y_{n} - y_{N}] - L| < ε/2.

Let us use the following relation

x_{n}/y_{n} - L = [x_{N} - Ly_{N}]/y_{n} + (1 - y_{N}/y_{n})([x_{n} - x_{N}]/[y_{n} - y_{N}] - L)

that leads to

|x_{n}/y_{n} - L| ≤ |[x_{N} - Ly_{N}]/y_{n}| + |[x_{n} - x_{N}]/[y_{n} - y_{N}] - L|.

The first expression on rhs is less than **ε/2** for **n > N' > N** because **y _{n} → ∞**. The second one is less than

|x_{n}/y_{n} - L| < ε.

what should be shown. On the other hand, if the limit

[x_{n} - x_{n-1}]/[y_{n} - y_{n-1}] = +∞

then

x_{n} - x_{n-1} > y_{n} - y_{n-1}

it can be only if

x_{n} → +∞.

Moreover, terms of sequence **{x _{n}}** increases with