Summation formulasnXi =11 = 1 + 1 + · · · + 1 + 1| {z }n terms= nnXi =1i = 1 + 2 + · · · + (n − 1) + n =12n(n + 1) =12n2+ O(n)nXi =1i2= 12+ 22+ · · · + (n − 1)2+ n2=13n(n +12)(n + 1) =13n3+ O(n2)nXi =1i3= 13+ 23+ · · · + (n − 1)3+ n3=14n2(n + 1)2=14n4+ O(n3)nXi =1im=1m+1nm+1+ O(nm)For more information, seehttp://www.math.rutgers.edu/~erowland/sumsofpowers.htmlDefinite integral factsZbacf (x) dx = cZbaf (x) dxZba(f (x) + g(x)) dx =Zbaf (x) dx +Zbag(x) dxZbac dx = c(b − a)If f (x) ≤ g (x) for all a ≤ x ≤ b, thenZbaf (x) dx ≤Zbag(x) dxIf m ≤ f (x ) ≤ M for all a ≤ x ≤ b, thenm(b − a) ≤Zbaf (x) dx ≤ M(b − a)Zcaf (x) dx +Zbcf (x) dx =Zbaf (x) dxZbaf (x) dx = −Zabf (x)
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