Euler's summation formula : Number Theory #Euler's #summation #formula #: #Number #Theory
Euler's summation formula : Statement : If f(x) has continuous derivative f’ on [a,b] where 0 < a< b , then 𝜮 a<n ≤ b f(n) = ∫ f(t)dt + ∫ (t – [t] ) dt +f(b)( [b] – b) -f(a) ( [a] -a ). Proof : Suppose f(x) has continuous derivative and f’(x) is in the closed interval [a,b] where 0 < a < b. Let [a] =m and [b] = k. Then 𝜮 a<n ≤ b f(n) = 𝜮 m<n ≤ k f(n) = 𝜮 f(n) …(i) | where in third sigma n is from m+1 to k | Suppose (n-1) and n are two integers in [a,b] and t lies between (n-1) and n. Then ∫ n-1 n [t] f’(t) dt = ∫ (n-1) f’(t) dt ...