[Paper Review] Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum

Leonhard Euler, Alexander Aycock|arXiv (Cornell University)|May 24, 2019

Historical and Literary Studies51 citations

TL;DR

Euler presents a method to transform and sum series using differential/difference calculus, notably via substitutions like x = y/(1+y), linking sums to iterated differences and enabling finite and infinite series summation.

ABSTRACT

This is the translation of Euler's Latin textbook Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum (second volume) into English.

Motivation & Objective

• Motivate and justify the use of differential calculus in analysis and series theory.
• Introduce auxiliary algebraic tools and the difference operator to transform series.
• Show how substitutions yield transformed series whose sums are expressible in closed form when differences stabilize.
• Illustrate finite and divergent series handling through transformed representations.

Proposed method

• Define a general power-series S = a x + b x^2 + c x^3 + … and substitute x = y/(1+y) to transform into a series with coefficients involving first, second, etc. differences of the sequence a, b, c, ….
• Express transformed sums as S = a y/(1−x) + Δa x^2/(1−x)^2 + Δ^2 a x^3/(1−x)^3 + … where Δ denotes forward differences.
• Demonstrate that when differences eventually become constant, the transformed series terminates and gives a finite closed form.
• Discuss alternating signs by taking x = −t and show analogous transformations.
• Apply the method to finite series by subtracting the tail’s transformed sum from the infinite transformed sum.
• Offer numerous explicit examples (e.g., arithmetic, squares, cubes, geometric, harmonic-like series) to illustrate convergence behavior and summation results.

Experimental results

Research questions

• RQ1 Can iterated forward differences of coefficient sequences yield closed-form sums for transformed series?
• RQ2 When can the substitution-based transformation produce finite expressions for otherwise infinite or divergent series?
• RQ3 How does changing the sign pattern (alternating series) affect the transformed representation and convergence?
• RQ4 How can transformations relate different series sums to each other and yield approximations for slow or non-convergent cases?

Key findings

• A general transformation maps a series into a sum of terms with powers of x/(1−x) multiplied by iterated differences, enabling sum extraction when differences stabilize.
• For series with constant first or higher-order differences, the sum can be written in finite closed form using the transformed coefficients (e.g., Δa, Δ^2a).
• Alternating-series can be transformed with x = 1/(1+x) to obtain rapidly converging representations and approximate sums even for divergent originals.
• The method works well for slowly converging or divergent series by producing faster-converging transformed series, and it provides a framework to derive known constants and functions through specific substitutions.
• Euler provides explicit worked examples (e.g., 1 − 1 + 1 − 1 …, 1 − 2 + 3 − 4 + …, series of squares, cubes, and hypergeometric-type progressions) illustrating the technique and its limits.
• The approach also yields relations between series and their tails, allowing finite sums to be obtained by subtracting tail-transformed sums from infinite-transformed sums.