Book: De seriebus divergentibus

Overview
Leonhard Euler's De seriebus divergentibus (1760) is a probing, often provocative examination of infinite series whose terms do not tend to zero and therefore fail to converge in the usual sense. Euler treats these "divergent" series not as meaningless formalities but as objects that can yield useful and consistent numerical values under careful manipulation. He approaches them with the same algebraic and analytic inventiveness that marks his other work, showing how formal operations and analytic continuation can produce finite quantities that fit into broader computational schemes.
The treatise occupies a transitional place between classical calculus and later, more rigorous summability theory. Euler's arguments are guided by intuition and formal consistency: if a manipulative procedure gives stable results across different equivalent transformations, that result deserves a name and further study. His examples range from simple alternating series to rapidly divergent power and factorial series, and he develops methods for assigning sums that anticipate techniques formalized much later.

Main ideas and methods
Central to Euler's approach is the idea that divergent series can be handled by extending algebraic identities and generating functions outside their radius of convergence. He uses termwise manipulation, re-expansion of known analytic functions, and transformation of series by finite differences to associate finite values with series that traditional convergence rejects. Euler often converts a problematic series into an expression involving a function defined by a convergent series or closed form, then evaluates that expression at parameter values for which the original series diverges.
Euler also exploits symmetries and functional relations to justify assignments. If a divergent series appears naturally as the analytic continuation or limiting case of a family of convergent series, the value obtained by continuation is taken to be the appropriate "sum." He applies algebraic operations, addition, subtraction, multiplication by polynomials, and term-by-term differentiation or integration, treating series formally while checking for consistency across different manipulative routes.

Key examples and results
Euler presents and defends several striking evaluations that have become emblematic of his treatment of divergence. The series 1 − 1 + 1 − 1 + … is assigned the value 1/2 by a reasoning that averages the two obvious partial-sum limits, and similar manipulations give 1 − 2 + 3 − 4 + … the value 1/4. Perhaps most famous is his assignment of 1 + 2 + 3 + 4 + … to the value −1/12, obtained via analytic continuation techniques closely related to later evaluations of the zeta function at negative integers. Euler also treats factorially divergent expansions and the formal inversion of power series, showing how apparently nonsensical expansions can be given coherent meanings when interpreted through generating functions.
Throughout these examples, Euler stresses consistency: different analytic continuations or equivalent algebraic rearrangements that lead to the same value strengthen the case for that assignment. He does not claim that divergent series converge in the ordinary sense, but he insists that useful and reproducible numerical values can often be attached to them.

Reception and legacy
Contemporary reaction to Euler's methods was mixed, with some mathematicians impressed by the utility of his results and others uneasy about the lack of rigorous foundations. The subject matured in the 19th and early 20th centuries through the development of summability methods, Abel and Cesàro summation, analytic continuation of Dirichlet series, and later Borel summation, all of which provided frameworks that vindicate many of Euler's claims. His bold formalism paved the way for systematic theories showing when and why such assignments are legitimate.
Euler's daring treatment of divergent series has had lasting influence beyond pure mathematics. Regularization techniques that echo his ideas are standard in theoretical physics, notably in quantum field theory and string theory, where divergent expressions are routinely assigned finite values that match experiment. De seriebus divergentibus stands as a striking example of mathematical creativity: it reshaped expectations about infinity and continuity and anticipated methods that became central to analysis and mathematical physics.