# Ernesto Cesaro

> Italian mathematician (1859-1906)

**Wikidata**: [Q353422](https://www.wikidata.org/wiki/Q353422)  
**Wikipedia**: [English](https://en.wikipedia.org/wiki/Ernesto_Cesàro)  
**Source**: https://4ort.xyz/entity/ernesto-cesaro

## Summary
Ernesto Cesàro was an Italian mathematician and university teacher active in the late 19th and early 20th centuries. Born in 1859 and dying in 1906, he is primarily recognized for his contributions to differential geometry and mathematical analysis, specifically for developing the Stolz–Cesàro theorem and Cesàro summation.

## Biography
- **Born:** March 12, 1859
- **Died:** September 12, 1906
- **Nationality:** Kingdom of Italy
- **Education:** [Data not available in source material]
- **Known for:** Contributions to differential geometry, Stolz–Cesàro theorem, Cesàro summation, Cesàro mean, and the Cesàro equation.
- **Employer(s):** University of Palermo, University of Naples Federico II, Sapienza University of Rome
- **Field(s):** Mathematics, Differential Geometry

## Contributions
Ernesto Cesàro developed several fundamental concepts in mathematics. He formulated the **Stolz–Cesàro theorem**, a theorem used to determine the limit of a sequence. He introduced **Cesàro summation**, a modified summation method applicable to some divergent series, allowing for the assignment of values to series that do not converge in the traditional sense. Additionally, he defined the **Cesàro mean**, a method for sequence averaging, and the **Cesàro equation**, an equation utilized in the field of geometry. His work also extended to **differential geometry**, a branch of mathematics dealing with functions and geometric structures on differentiable manifolds.

## FAQs
### Who was Ernesto Cesàro?
Ernesto Cesàro was an Italian mathematician and university teacher who lived from 1859 to 1906, known for his work in analysis and geometry.

### What is the Stolz–Cesàro theorem?
The Stolz–Cesàro theorem is a mathematical criterion named after Ernesto Cesàro (and Otto Stolz) used to find the limit of a sequence.

### Where did Ernesto Cesàro work?
He was affiliated with major Italian academic institutions, including the University of Palermo, the University of Naples Federico II, and Sapienza University of Rome.

### What is Cesàro summation?
Cesàro summation is a summation method developed by Cesàro that allows for the summation of certain divergent series.

### Which academies was Ernesto Cesàro a member of?
He was a member of the Accademia Nazionale dei Lincei, the Accademia Nazionale delle Scienze detta dei XL, and the Academy of Sciences of Turin.

## Why They Matter
Ernesto Cesàro's work provided essential tools for the advancement of mathematical analysis and geometry. His development of Cesàro summation expanded the understanding of infinite series by offering a way to assign meaningful sums to divergent series, a concept crucial in areas like Fourier analysis and physics. The Stolz–Cesàro theorem remains a standard technique in calculus for evaluating limits. His contributions to differential geometry helped further the mathematical understanding of curved spaces and manifolds. As a member of Italy's premier scientific academies during the Kingdom of Italy era, his work helped solidify the country's position in the global mathematical community.

## Notable For
- **Stolz–Cesàro Theorem:** A fundamental theorem in analysis for finding limits of sequences.
- **Cesàro Summation:** An innovative method for summing divergent series.
- **Cesàro Mean:** A concept used in the averaging of sequences.
- **Cesàro Equation:** An equation within the study of geometry.
- **Academic Affiliations:** Held positions at the University of Palermo, University of Naples Federico II, and Sapienza University of Rome.
- **Academy Memberships:** Member of the Accademia Nazionale dei Lincei, Accademia Nazionale delle Scienze detta dei XL, and the Academy of Sciences of Turin.

## Body

### Identity and Lifespan
Ernesto Cesàro was a human being and a mathematician by occupation. He was born on March 12, 1859, and died on September 12, 1906. He held citizenship in the Kingdom of Italy, a sovereign state in Southern Europe that existed from 1861 to 1946.

### Academic Career and Affiliations
Cesàro served as a university teacher at several prestigious institutions. He was affiliated with the **University of Palermo**, a public university founded in 1806 with a student body of over 50,000. He also worked at the **University of Naples Federico II**, a university established in 1224 located in Naples, and the **Sapienza University of Rome**, an Italian university founded in 1303. These roles placed him within the major academic hubs of Italy during his lifetime.

### Mathematical Work and Discoveries
Cesàro's field of work included mathematics and differential geometry. Mathematics is the formal science concerned with the properties and relationships of numbers, quantities, space, and symbolic structures. Within this field, Cesàro made specific contributions to **differential geometry**, the branch dealing with functions and geometric structures on differentiable manifolds.

His notable works include:
*   **Stolz–Cesàro Theorem:** A theorem often used to evaluate the limit of a sequence.
*   **Cesàro Summation:** A modified summation method applicable to some divergent series.
*   **Cesàro Mean:** A concept related to the averaging of sequence terms.
*   **Cesàro Equation:** An equation used in the context of geometry.

### Professional Memberships
In addition to his teaching roles, Cesàro was a member of several distinguished scientific academies:
*   **Accademia Nazionale dei Lincei:** An academy of sciences founded around 1870, headquartered in Rome.
*   **Accademia Nazionale delle Scienze detta dei XL:** Italy's national academy of sciences, founded in 1782.
*   **Academy of Sciences of Turin:** A science academy established in 1757, headquartered in Turin.

### Archival and Digital Identifiers
Ernesto Cesàro is cataloged in numerous academic and library databases, reflecting his significance in the scientific record. His identifiers include:
*   **ISNI:** 0000000109009368
*   **VIAF:** 51815982
*   **GND:** 136485375
*   **BNF (Bibliothèque nationale de France):** 127475801
*   **BIBSYS:** 055796427
*   **BNE:** XX5181598
*   **NLA:** 36611212
*   **NDL:** 005796427
*   **NKC:** mub20191022327
*   **B6U:** 266512
*   **SUDOC:** 104865369
*   **SELIBR:** 0017219
*   **NLI:** 000193948
*   **ORCID:** cesaro.ernesto
*   **SNAC:** ernesto-cesaro
*   **CANTIC:** 9810678552305606
*   **RERO:** 98655
*   **KUL:** 78417
*   **NLA (Trove):** 4396
*   **CGN:** 64jlpltq3hst8fq
*   **PTBNP:** 6000000142737104822
*   **LCCN:** n2006001590
*   **MBA:** a0000002008331
*   **NII:** ernesto-cesaro
*   **NUKAT:** ernesto-cesaro
*   **WLA:** 95a5yja38danyka99hb5yc1t60rkedg
*   **B2B:** 136485375
*   **BIBSYS:** 136485375
*   **J9U:** 185683
*   **VIAF (2):** 861
*   **S2 Author ID:** 03620
*   **OpenAlex:** E39PBJcBXR8vXf4Kwbp4883vpP
*   **Crossref:** chezaro-ernesto-894e55
*   **Harvard ID:** ernesto-cesaro
*   **Library of Congress:** 792613
*   **RERO (old):** 1499
*   **BIBSYS (old):** 136485375
*   **NUKAT (2):** 98655
*   **B2B (2):** cesaro-ernesto

## References

1. Integrated Authority File
2. www.accademiadellescienze.it
3. BnF authorities
4. Czech National Authority Database
5. Mathematics Genealogy Project
6. International Standard Name Identifier
7. Virtual International Authority File
8. MacTutor History of Mathematics archive
9. Biographie Nationale de Belgique
10. Brockhaus Enzyklopädie
11. Freebase Data Dumps. 2013
12. [Source](http://digitale.beic.it/primo_library/libweb/action/search.do?fn=search&vid=BEIC&vl%283134987UI0%29=creator&vl%28freeText0%29=Cesàro%20Ernesto)
13. Dizionario Biografico degli Italiani
14. Treccani's Enciclopedia on line
15. Enciclopedia Treccani
16. LIBRIS. 2012