In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. It is named after mathematicians Otto Stolz and Ernesto Cesàro, who stated and proved it for the first time. The Stolz–Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital's rule for sequences.
Statement of the theorem for the */∞ case Let ( a n ) n ≥ 1 {\displaystyle (a_{n})_{n\geq 1}} and ( b n ) n ≥ 1 {\displaystyle (b_{n})_{n\geq 1}} be two sequences of real numbers. Assume that ( b n ) n ≥ 1 {\displaystyle (b_{n})_{n\geq 1}} is a strictly monotonic and divergent sequence (i.e. strictly increasing and approaching + ∞ {\displaystyle +\infty } , or strictly decreasing and approaching − ∞ {\displaystyle -\infty } ) and the following limit exists:
lim n → ∞ a n + 1 − a n b n + 1 − b n = l . {\displaystyle \lim _{n\to \infty }{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}=l.\ }
Then, the limit
lim n → ∞ a n b n = l . {\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}=l.\ }
Statement of the theorem for the 0/0 case Let ( a n ) n ≥ 1 {\displaystyle (a_{n})_{n\geq 1}} and ( b n ) n ≥ 1 {\displaystyle (b_{n})_{n\geq 1}} be two sequences of real numbers. Assume now that ( a n ) → 0 {\displaystyle (a_{n})\to 0} and ( b n ) → 0 {\displaystyle (b_{n})\to 0} while ( b n ) n ≥ 1 {\displaystyle (b_{n})_{n\geq 1}} is strictly decreasing. If
lim n → ∞ a n + 1 − a n b n + 1 − b n = l , {\displaystyle \lim _{n\to \infty }{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}=l,\ }
then
lim n → ∞ a n b n = l . {\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}=l.\ }
Proofs
Proof of the theorem for the */∞ case Case 1: suppose ( b n ) {\displaystyle (b_{n})} strictly increasing and divergent to + ∞ {\displaystyle +\infty } , and − ∞ < l < ∞ {\displaystyle -\infty <l<\infty } . By hypothesis, we have that for all ϵ / 2 > 0 {\displaystyle \epsilon /2>0} there exists ν > 0 {\displaystyle \nu >0} such that ∀ n > ν {\displaystyle \forall n>\nu }
… excerpt ends here. Continue reading the full article.
