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Business Mathematics and Basic Statistics · Ch 7 — Arithmetic and Geometric Progressions

Some Important Series

6

Some Important Series

Beyond AP and GP, three specific series of natural numbers are important enough in business mathematics and statistics (they reappear in later work on means and moments) to be stated here by name, using the summation symbol ∑\sum (sigma), where ∑i=1ni\displaystyle\sum_{i=1}^{n} i means "add up ii for every whole-number value of ii from 11 to nn":

Note

The Three Standard Summation Formulas

  • Sum of the first nn natural numbers: ∑i=1ni=1+2+3+⋯+n=n(n+1)2\sum_{i=1}^{n} i = 1+2+3+\cdots+n = \frac{n(n+1)}{2}
  • Sum of the squares of the first nn natural numbers: ∑i=1ni2=12+22+32+⋯+n2=n(n+1)(2n+1)6\sum_{i=1}^{n} i^2 = 1^2+2^2+3^2+\cdots+n^2 = \frac{n(n+1)(2n+1)}{6}
  • Sum of the cubes of the first nn natural numbers: ∑i=1ni3=13+23+33+⋯+n3=(n(n+1)2)2\sum_{i=1}^{n} i^3 = 1^3+2^3+3^3+\cdots+n^3 = \left(\frac{n(n+1)}{2}\right)^2

Notice the sum of the first nn natural numbers is itself exactly the AP sum formula (§7) applied to the AP 1,2,3,…,n1,2,3,\ldots,n with a=1,d=1a=1,d=1 — these three results connect directly back to what this chapter has already built. …

Definition 5Summation notation, Σ

∑i=1nf(i)\sum_{i=1}^{n} f(i) means the sum f(1)+f(2)+⋯+f(n)f(1)+f(2)+\cdots+f(n) — add the expression for every whole-number value of …

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