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September 8

Σ and ʃ

How are summation and integration are related? That is to say, if i can only perform a sum of a function from a to b how can I find the area, and if I can only integrate a function along [a, b] how can I find the sum of it (or it's sequence)? 24.92.78.167 (talk) 02:34, 8 September 2010 (UTC)[reply]

Two relations are the definition of the integral via Riemann integral#Riemann sums and the Integral test for convergence of series. -- 58.147.53.113 (talk) 05:21, 8 September 2010 (UTC)[reply]

There are also the Euler–Maclaurin formula which works both ways, and methods like Simpson's rule and Gaussian quadrature that can be used to approximate an integral using a sum. -- Meni Rosenfeld (talk) 08:06, 8 September 2010 (UTC)[reply]

They are related by Taylor's theorem. Write Δ for the difference operator (Δf (x) = f(x+1)-f(x) ) and write D for differentiation, so D = 1/ʃ and Δ = 1 / Σ. The classic game to play with Σ, ʃ, Δ and D is as follows: you can interpret Taylor as saying Δ = exp(D)-1. So Σ = 1/(exp(D)-1) = (1/D)(D / (exp(D)-1)) = ʃ (1 - (1/2) D + (1/12)D^2 - (1/720)D^4 ...) = ʃ + 1/2 + (1/12)D - (1/720)D^3 + ... This can actually be made rigourous, at the expense of some error terms and conditions on the functions f to which it is applied: you end up with the Euler-Maclaurin summation formula mentioned by Meni. The coefficients in the power series are closely related to the Bernoulli numbers. Tinfoilcat (talk) 17:59, 8 September 2010 (UTC)[reply]