Stolz-Cesàro Theorem/Corollary

Corollary to Stolz-Cesàro Theorem

Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences in $\R$ such that $\sequence {b_n}$ is strictly increasing and $\ds \lim_{n \mathop \to \infty} b_n = \infty$.

If:

$\ds \lim_{n \mathop \to \infty} \frac {a_n - a_{n - 1} } {b_n - b_{n - 1} } = L \in \R$

then also:

$\ds \lim_{n \mathop \to \infty} \frac {a_n} {b_n} = L$

Proof

Define the following sequences:

$x_1 = a_1$, $x_n = a_n - a_{n - 1}$

$y_1 = b_1$, $y_n = b_n - b_{n - 1}$

It follows that:

$\ds \sum_{i \mathop = 1}^n x_i = a_n$

and:

$\ds \sum_{i \mathop = 1}^n y_i = b_n$

From above follows:

$\ds \lim_{n \mathop \to \infty} \frac {x_n} {y_n} = \lim_{n \mathop \to \infty} \frac {a_n - a_{n - 1} } {b_n - b_{n - 1} } = L$

From the definition of divergent sequences there exists $N \in \N$ such that

$b_n$ is positive for all $n > N$

From the general Stolz-Cesàro Theorem follows that

$\ds \lim_{n \mathop \to \infty} \frac {a_n} {b_n} = \lim_{n \mathop \to \infty} \frac {\sum_{i \mathop = 1}^n x_i} {\sum_{i \mathop = 1}^n y_i} = L$

Hence the result.

$\blacksquare$

Remarks

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• The theorem also holds if $\sequence {b_n}$ is strictly monotone and $\ds \lim_{n \mathop \to \infty} b_n = \pm \infty$.
• Just as the general Stolz-Cesàro Theorem, the corollary also a version using limit inferior and limit superior. In that case the limit $L$ can be a real number or $\pm \infty$.
• This form of Stolz-Cesàro Theorem is most commonly found in books.
• The theorem can be considered as a special case of L'Hopital's Rule for sequences.

Source of Name

This entry was named for Ernesto Cesàro and Otto Stolz.