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| A '''sum''' is the addition of two or more ]s (see ]). A sum of several terms is usually written using plus symbols (+). A sum with many terms is often written using a capital sigma, which is defined as: | A '''sum''' is the ] of two or more ]s (see ]). A sum of several terms is usually written using plus symbols (+). A sum with many terms is often written using a capital sigma, which is defined as: | ||
| ''b'' | ''b'' | ||
Revision as of 00:54, 25 August 2002
A sum is the addition of two or more numbers (see arithmetic). A sum of several terms is usually written using plus symbols (+). A sum with many terms is often written using a capital sigma, which is defined as:
b ∑ f(i) = f(a) + f(a+1) + f(a+2) + ... + f(b-1) + f(b) i=a
When b is replaced with the ∞ symbol, the sum is an infinite series. It has an infinite number of terms, and represents the limit of the sum of the first n terms, as n grows without bound.
The following are useful identities:
n ∑ i = n(n+1)/2 i=1
n ∑ i = (2n+3n+n)/6 i=0
n ∑ x = (x -1) / (x-1) i=0
∞ ∑ x = 1 / (1-x) i=0
n-1 / i \ / n \ ∑ | | = | | (see binomial coefficient) i=0 \ k / \ k+1 /
The following are useful approximations (using theta notation):
n ∑ i = Θ(n) for every real constant c ≠ -1. i=1
n ∑ 1/i = Θ(log(n)) i=1
n ∑ c = Θ(c) for every real constant c. i=1
n ∑ log(i) = Θ(n log(n)) for every real constant c ≥ 0. i=1
n ∑ log(i) i = Θ(n log(n)) for all real constants c ≥ 0 and d ≥ 0. i=1
n ∑ log(i) i b = Θ(n log(n) b) for all real constants c ≥ 0, d ≥ 0 and b > 1. i=1
See also: power series, formal power series Taylor series, Fourier series, Laurent series,