Summation of Series


In sequences and series, the summation of series refers to the process of finding the sum of terms in a sequence up to a certain number of terms or to infinity. Various techniques and formulas exist to calculate the sum of specific types of series.


The sigma () notation is commonly used to represent the summation of series:



  • ak represents the terms of the series for k ranging from 1 to n.
  • n is the upper limit of summation.

Types of Series and Formulas:

  1. Arithmetic Series:

    • Sum of an arithmetic sequence.
    • Formula: Sn=n2×[2a+(n1)×d], where Sn is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.
  2. Geometric Series:

    • Sum of a geometric sequence.
    • Formula: Sn=a×(1rn)1r, where Sn is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
  3. Harmonic Series:

    • Sum of a harmonic sequence.
    • Harmonic series diverges to infinity: n=11n.

Properties and Techniques:

Summation Properties:

  1. Additive Property:

    • k=1n[ak+bk]=k=1nak+k=1nbk
  2. Multiplicative Property:

    • k=1ncak=ck=1nak
  3. Changing the Index:

    • k=mnak=j=mn+pajp where p is an integer.
  1. Partial Sums:

    • Calculating the sum of a specific number of terms in a series.
    • Useful for finding finite sums.
  2. Infinite Series:

    • Series that extend to an infinite number of terms.
    • Determining convergence or divergence is essential.
  3. Summation Rules:

    • Rearranging terms, splitting series, using telescoping series, etc., to simplify calculations.
  4. Convergence Tests:

    • Various tests like the divergence test, ratio test, integral test, etc., to determine if an infinite series converges or diverges.


  • Mathematics: Calculating areas, volumes, and understanding limits.
  • Physics: Modelling continuous phenomena, calculating total energy, etc.
  • Engineering: Signal processing, control systems, etc.
  • Finance: Compound interest, annuities, etc.