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Part of ALG-03 — Sequences & Series (AP, GP, Special Series)

Special Series Summation

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Standard sums (memorize these):

Series	Formula
$sum_{k=1}^{n}$ k	n $\frac{n+1}{2}$
$sum_{k=1}^{n}$ $k^2$	n(n+1) $\frac{2n+1}{6}$
$sum_{k=1}^{n}$ $k^3$	[n $\frac{n+1}{2}$ ]^2
$sum_{k=1}^{n}$ $k^4$	n(n+1)(2n+1) $\frac{3n^2+3n-1}{30}$
$sum_{k=1}^{n}$ (2k-1)	$n^2$
$sum_{k=1}^{n}$ 2k	n(n+1)
$sum_{k=1}^{n}$ k(k+1)	n(n+1) $\frac{n+2}{3}$
$sum_{k=1}^{n}$ k(k+1)(k+2)	n(n+1)(n+2) $\frac{n+3}{4}$

Beautiful identity: sum( $k^3$ ) = [sum(k)]^2. The sum of cubes equals the square of the sum of natural numbers.

General pattern: sum of k(k+1)(k+2)...(k+m-1) = n(n+1)(n+2)...(n+m) / (m+1). This generalizes the first few formulas.

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