Stolz–Cesàro theorem

The set of pride, stolzsche limit theorem or set of Pride Cesàro is of limits in mathematics. It is named after the Austrian mathematician Otto Stolz (1842-1905) and the Italian mathematician Ernesto Cesàro ( 1859-1906 ).

Set

Are and consequences in with

And the limit exists

Then:

Proof of the second case

Following the adoption of convergence of the difference quotient with a limit exists for each one, so that is all the difference quotient to the index in the area. So there is for each a with

For true.

Summing up these relations after from to, we obtain the equation

Thus applies to the quotient of the followers

The first term of the right-hand side converges to zero as the sequence grows unbounded. For the same reason, the second term converges to. Due to the monotonicity of the sequence applies for the third summand

You can now find, such that for all in the first two summands the difference to the threshold is limited by, for all, we obtain the estimate of a

Thus converges to the sequence of quotients.

To reverse

The converse of the above theorem is false in general. Looking at the two episodes

Then we have. However, the result has no limit.

Generalization

Given two more episodes and such that and. It should also be strictly monotonically increasing and unbounded.

From

Then follows

The above requirements are met by

• The harmonic series, and
• Of each row, the links have a positive threshold value, as, i.e., or even
• Each row, grow their limbs even as, ie.

Comments

A special case is the Cauchy's limit theorem.

In a way, the set of Pride an equivalent for calculating the limit values ​​for consequences to the rule of L'Hospital for limit calculation for differentiable functions dar.