Series, Sigma Notation & Infinite Sums

From sequence to series

A series is what you get by adding the terms of a sequence. The sum of the first n terms is the n-th partial sum, written Sₙ. So if the sequence is 3, 7, 11, 15, then S₄ = 3 + 7 + 11 + 15 = 36.

Summation notation (the Greek capital sigma, Σ) packages a long sum compactly. The expression below Σ is a starting index, the number on top is the ending index, and the formula to the right tells you what to add. You substitute each index value, then total the results.

Read and expand a sigma sum:

  sum from k=1 to 4 of (2k + 1)

Substitute k = 1, 2, 3, 4:
  k=1: 2(1)+1 = 3
  k=2: 2(2)+1 = 5
  k=3: 2(3)+1 = 7
  k=4: 2(4)+1 = 9

Add:  3 + 5 + 7 + 9 = 24

Partial-sum formulas

For an arithmetic series there is a beautiful shortcut. Pair the first term with the last, the second with the second-to-last, and so on: each pair sums to (a₁ + aₙ). There are n terms, so n/2 such pairs, giving Sₙ = n(a₁ + aₙ)/2 — the average of the first and last term, times how many terms.

Sum 1 + 2 + 3 + ... + 100  (arithmetic, d = 1)

a_1 = 1,  a_n = 100,  n = 100

S_n = n(a_1 + a_n)/2
S_100 = 100(1 + 100)/2
      = 100(101)/2
      = 10100/2
      = 5050

For a geometric series the partial sum is Sₙ = a₁(1 − rⁿ)/(1 − r), valid whenever the common ratio r ≠ 1. For example, 2 + 6 + 18 + 54 has a₁ = 2, r = 3, n = 4, so S₄ = 2(1 − 3⁴)/(1 − 3) = 2(1 − 81)/(−2) = 2(−80)/(−2) = 80.

When infinity adds to a finite number

Here is the surprise. If a geometric ratio satisfies |r| < 1, the terms shrink toward zero fast enough that an infinite geometric series adds up to a finite total: S = a₁/(1 − r). Look at the partial-sum formula — when |r| < 1, the rⁿ piece fades to 0 as n grows, leaving exactly a₁/(1 − r).

Infinite series:  1 + 1/2 + 1/4 + 1/8 + ...

a_1 = 1,   r = 1/2   (and |1/2| < 1, so it converges)

S = a_1 / (1 - r)
  = 1 / (1 - 1/2)
  = 1 / (1/2)
  = 2

The endless sum settles on exactly 2.