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This page contains the Stolz's theorem.
This theorem is very useful when one can find a limit of expressions xn/yn that are of indeterminate form i. e. ∞/∞.
Let yn → +∞. Moreover, starting from some point yn is increasing with n i. e.
then
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
and
It means that for any n > N all fractions
are between L - ε/2 and L + ε/2. Because from some point
among them is also the fraction (obtained as a sum of all nominators and denominators)
and for n > N is
Let us use the following relation
that leads to
The first expression on rhs is less than ε/2 for n > N' > N because yn → ∞. The second one is less than ε/2 as well. Thus one has
what should be shown. On the other hand, if the limit
then
it can be only if
Moreover, terms of sequence {xn} increases with n.
