Stolz-Cesàro Theorem/Corollary
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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.