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Canada ((ARW)) Talk

		The Original Barnstar
		{{{1}}}

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Ducks

Female Mallard

Scientific classification

Subfamilies

Dendrocygninae
Oxyurinae
Anatinae
Merginae

The following are useful identities:

$\sum_{i=1}^{n} i = \frac {n(n+1)}{2}$

$\sum_{i=1}^n m = mn$

$\sum_{k=i}^nk = \frac{(n-i+1)(n+i)}{2}$

$\sum_{i=1}^{n} (2i - 1) = n^2;$

$\sum_{i=0}^{n} i^{2} = \frac{n(n+1)(2n+1)}{6};$

$\sum_{i=0}^{n} i^{3} = \left(\frac{n(n+1)}{2}\right)^{2};$

$\sum_{i=N_1}^{N_2} x^{i} = \frac{x^{N_2+1}-x^{N_1}}{x-1}$ (see geometric series);

$\sum_{i=0}^{n} x^{i} = \frac{1-x^{n+1}}{1-x}$ (special case of the above where

N 1 = 0

)

$\sum_{i=0}^{\infty} x^{i} = \frac{1}{1-x};$ (special case of the above, $\lim_{n\to\infty}$ and

| x | < 1

);

$\sum_{i=0}^{n} {n \choose i} = 2^{n}$

$\sum_{i=0}^{n-1} {i \choose k} = {n \choose k+1}.$

$\left(\sum_i a_i\right)\left(\sum_j b_j\right) = \sum_i\sum_j a_ib_j$

${\left(\sum_i a_i\right)}^2 = 2\sum_i\sum_{j<i} a_ia_j + \sum_i a_i^2$

In general, the sum of the first n mth powers is

$\sum_{i=0}^n i^m = \frac{(n+1)^{m+1}}{m+1} + \sum_{k=1}^m\frac{B_k}{m-k+1}{m\choose k}(n+1)^{m-k+1},$

where $B k$ is the kth Bernoulli number.