Math 2.0

Sequences, series, summability Convergence tests Theorem

Abel's summation formula

Builds on Continuously differentiable functionRiemann integralIntegration by parts

Statement

Let (an)n≥1(a_n)_{n \ge 1} be a sequence of complex numbers, and for real tt put A(t)=∑1≤n≤tanA(t) = \sum_{1 \le n \le t} a_n. Let x≥1x \ge 1 be a real number, and let φ\varphi be a continuously differentiable complex-valued function on [1,x][1, x]. Then ∑1≤n≤xanφ(n)=A(x)φ(x)−∫1xA(t)φ′(t) dt.\sum_{1 \le n \le x} a_n \varphi(n) = A(x) \varphi(x) - \int_1^x A(t) \varphi'(t) \, dt.

Remarks

The function AA is constant on each interval [k,k+1)[k, k+1), so the integral splits as ∑kA(k)(φ(k+1)−φ(k))\sum_k A(k) (\varphi(k+1) - \varphi(k)) plus a last piece up to xx. Rearranging this finite sum gives the left side; this is summation by parts. The formula is the discrete analogue of integration by parts, and it is integration by parts for the Riemann–Stieltjes integral ∫φ dA\int \varphi \, dA. Example: with an=1a_n = 1 and φ(t)=1/t\varphi(t) = 1/t, one has A(t)=⌊t⌋A(t) = \floor{t}, and ∑1≤n≤x1n=⌊x⌋x+∫1x⌊t⌋t2 dt.\sum_{1 \le n \le x} \frac{1}{n} = \frac{\floor{x}}{x} + \int_1^x \frac{\floor{t}}{t^2} \, dt. Writing ⌊t⌋=t−{t}\floor{t} = t - \{t\} shows that the harmonic numbers minus the logarithm converge.

Going deeper

For a student

In analytic number theory one often knows a counting function A(x)A(x), such as the number of primes up to xx, and wants sums like ∑p≤x1/p\sum_{p \le x} 1/p. Abel's formula transfers estimates for AA into estimates for such weighted sums. It also extends the Riemann zeta function to a meromorphic function on Re⁡(s)>0\operatorname{Re}(s) > 0 with a single pole at s=1s = 1, by taking an=1a_n = 1 and φ(t)=t−s\varphi(t) = t^{-s}. The Dirichlet's test and Abel's test for series rest on the same summation by parts.

Reading path

To understand this page from the ground up, read these 36 pages in order. The foundations come first.

Show the path as a graph
Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementRiemann integral, definition, no evidence levelRiemann integraldefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementDifferentiable functions are continuous, statement, no evidence levelDifferentiable functi…statementDerivative function, definition, no evidence levelDerivative functiondefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementProduct rule, statement, no evidence levelProduct rulestatementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementHigher derivative, definition, no evidence levelHigher derivativedefinitionExtreme value theorem, statement, no evidence levelExtreme value theoremstatementContinuously differentiable function, definition, no evidence levelContinuously differen…definitionRolle's theorem, statement, no evidence levelRolle's theoremstatementMean value theorem, statement, no evidence levelMean value theoremstatementSecond fundamental theorem of calculus, statement, no evidence levelSecond fundamental th…statementIntegration by parts, statement, no evidence levelIntegration by partsstatementAbel's summation formula, statement, no evidence levelAbel's summation form…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Limit point of a set definition
  2. Local maximum definition
  3. Partition of an interval definition
  4. Sequence definition
  5. Limit of a function definition
  6. Local minimum definition
  7. Mesh of a partition definition
  8. Riemann sum definition
  9. Real sequence definition
  10. Subsequence definition
  11. Continuity at a point definition
  12. Derivative definition
  13. Riemann integrable function definition
  14. Bounded sequence definition
  15. Limit of a sequence definition
  16. Monotone sequence definition
  17. Continuous function definition
  18. Differentiable function definition
  19. Fermat's theorem on stationary points statement
  20. Riemann integral definition
  21. Convergent sequence definition
  22. Monotone convergence theorem for sequences statement
  23. Every sequence has a monotone subsequence statement
  24. Differentiable functions are continuous statement
  25. Derivative function definition
  26. Bolzano–Weierstrass theorem statement
  27. Product rule statement
  28. Boundedness theorem statement
  29. Higher derivative definition
  30. Extreme value theorem statement
  31. Continuously differentiable function definition
  32. Rolle's theorem statement
  33. Mean value theorem statement
  34. Second fundamental theorem of calculus statement
  35. Integration by parts statement
  36. Abel's summation formula statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Abel%27s_summation_formula
About this page
TagV5EPT2PX9Z
URIm2://V5EPT2PX9Z
Kindstatement, theorem
WritingW0 stub
MSC40A05, 11A25
StatusThis copy of the site has no build data. Run m2 status V5EPT2PX9Z.
UsesContinuously differentiable function, Riemann integral, Integration by parts
Same as
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.