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Mathematics and Statistics · Ch 4 — Sequences and Series

Special Series: Sums of Powers of Natural Numbers

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Special Series: Sums of Powers of Natural Numbers

Three standard series of the natural numbers appear constantly in statistics (they reappear in means, variance and moments), and the syllabus states them by name using the summation symbol ∑\sum (sigma), where ∑r=1nr\displaystyle\sum_{r=1}^{n} r means "add rr for every whole number rr from 11 to nn".

Note

The Three Standard Sums

  • First nn natural numbers: ∑r=1nr=1+2+⋯+n=n(n+1)2\sum_{r=1}^{n} r = 1 + 2 + \cdots + n = \frac{n(n+1)}{2}
  • Squares of the first nn natural numbers: ∑r=1nr2=12+22+⋯+n2=n(n+1)(2n+1)6\sum_{r=1}^{n} r^2 = 1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}
  • Cubes of the first nn natural numbers: ∑r=1nr3=13+23+⋯+n3=(n(n+1)2)2\sum_{r=1}^{n} r^3 = 1^3 + 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2

Notice the first sum is just the AP sum formula (§3) applied to 1,2,3,…,n1, 2, 3, \ldots, n (where a=1,d=1a = 1, d = 1), and the cube sum is the square of the first sum — the three results connect back to what this chapter already built.

Note

Linearity — Splitting a Sum

∑( p r2+q r+c )=p∑r2+q∑r+cn\sum (\,p\,r^2 + q\,r + c\,) = p\sum r^2 + q\sum r + cn: a sum of several parts can be split into separate standard sums, each evaluated by its own formula, then combined. (Here ∑r=1nc=cn\sum_{r=1}^{n} c = cn, since the constant cc is added nn times.) …

Definition 14Summation notation (Σ)

∑r=1nf(r)\sum_{r=1}^{n} f(r) means f(1)+f(2)+⋯+f(n)f(1) + f(2) + \cdots + f(n) — add the expression for every whole number $ …

Definition 15Standard power sums

∑r=n(n+1)2\sum r = \frac{n(n+1)}{2}, ∑r2=n(n+1)(2n+1)6\sum r^2 = \frac{n(n+1)(2n+1)}{6}, ∑r3=(n(n+1)2)2\sum r^3 = \left(\frac{n(n+1)}{2}\right)^2, …

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