Sigma Notation — Quick Reference

Σ Notation basics

  • Mathematicians use the sigma notation to denote a sum. The uppercase Greek letter Σ indicates a sum.
  • Notation consists of: summand, index, lower bound, and upper bound.
  • General form: ∑i=mnf(i)\sum_{i=m}^{n} f(i) reads as “the summation of f(i) from i = m to n,” where m≤nm\le n are integers and f(i)f(i) is a term (summand).

Components of sigma notation

  • Index: ii
  • Lower bound: mm
  • Upper bound: nn
  • Summand: f(i)f(i)
  • Expansion form: ∑i=mnf(i)=f(m)+f(m+1)+⋯+f(n)\sum_{i=m}^{n} f(i) = f(m) + f(m+1) + \cdots + f(n)

Examples of expanding sums

  • Example 1: ∑i=24(2i−3)=(2⋅2−3)+(2⋅3−3)+(2⋅4−3)=1+3+5=9\sum_{i=2}^{4} (2i - 3) = (2\cdot2 - 3) + (2\cdot3 - 3) + (2\cdot4 - 3) = 1 + 3 + 5 = 9
  • Example 2: ∑i=052i=1+2+4+8+16+32=63\sum_{i=0}^{5} 2^{i} = 1 + 2 + 4 + 8 + 16 + 32 = 63

Write each expression in sigma notation

  • 1 + 1/2 + 1/3 + 1/4 + \cdots + 1/100 = ∑i=11001i\sum_{i=1}^{100} \frac{1}{i}
  • -1 + 2 - 3 + 4 - 5 + 6 - 7 + \cdots - 25 = ∑i=125(−1)ii\sum_{i=1}^{25} (-1)^i i
  • a2 + a4 + a6 + a8 + \cdots + a20 = ∑<em>i=110a</em>2i\sum<em>{i=1}^{10} a</em>{2i}
  • 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 = ∑j=07(12)j\sum_{j=0}^{7} \left(\frac{1}{2}\right)^j

Seatwork / properties of sigma notation

  • Basic property: Sum of a linear combination
    ∑<em>i=mn[c f(i)+d g(i)]=c ∑</em>i=mnf(i)+d ∑i=mng(i)\sum<em>{i=m}^{n} [c\,f(i) + d\,g(i)] = c\,\sum</em>{i=m}^{n} f(i) + d\,\sum_{i=m}^{n} g(i)
  • Constant multiple rule: brought out of the summation.
  • Sum of a sequence: ∑<em>i=1na</em>i=a<em>1+a</em>2+⋯+an\sum<em>{i=1}^{n} a</em>i = a<em>1 + a</em>2 + \cdots + a_n
  • Common results:
    • ∑i=1ni=n(n+1)2\sum_{i=1}^{n} i = \frac{n(n+1)}{2}
    • ∑i=1ni2=n(n+1)(2n+1)6\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}

Key evaluated sums

  • Example: ∑i=130(4i−5)=1710\sum_{i=1}^{30} (4i - 5) = 1710
  • Example: ∑k=1991k(k+1)=99100\sum_{k=1}^{99} \frac{1}{k(k+1)} = \frac{99}{100}

More practice: quick reference conversions

  • Write the expression in sigma notation: 1 + 1/2 + 1/3 + \cdots + 1/100 → ∑i=11001i\sum_{i=1}^{100} \frac{1}{i}
  • Write the alternating sum: -1 + 2 - 3 + 4 - 5 + \cdots - 25 → ∑i=125(−1)ii\sum_{i=1}^{25} (-1)^i i
  • Even terms sum: a2 + a4 + a6 + a8 + a10 → ∑<em>i=15a</em>2i\sum<em>{i=1}^{5} a</em>{2i}

Quick formulas to remember

  • Sum of a finite sequence: ∑<em>i=1na</em>i=a<em>1+a</em>2+⋯+an\sum<em>{i=1}^{n} a</em>i = a<em>1 + a</em>2 + \cdots + a_n
  • Linearity and constants allow combining and scaling sums.
  • Common closed forms:
    • ∑i=1ni=n(n+1)2\sum_{i=1}^{n} i = \frac{n(n+1)}{2}
    • ∑i=1ni2=n(n+1)(2n+1)6\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}