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-1ⁿ [t] f’(t) dt = ∫ (n-1) f’(t) dt

                                   = (n-1) ∫ f’(t) dt

                                    = (n-1) [f(t)] _n-1ⁿ

                                    = (n-1) ( f(n) – f(n-1)

                                    = n f(n) – f(n) – (n-1) f(n-1)  … (ii)

Taking summation with n = m+1, m+2, … , k on both sides of (ii) , we get

    𝜮 ∫_n-1ⁿ [t] f’(t) dt  = 𝜮 (n f(n) – f(n) – (n-1) f(n-1) )

⇒ 𝜮 ∫_n-1ⁿ [t] f’(t) dt = k f(k) – m f(m) – 𝜮 f(n)

⇒                   𝜮 f(n) = k f(k) – m f(m) – 𝜮 ∫_n-1ⁿ [t] f’(t) dt

                                     = k f(k) – m f(m) –  ∫_a^b [t] f’(t) dt

                                     = k f(b) – m f(a) –  ∫_a^b [t] f’(t) dt  … (iii)

Now ∫_a^b f(t) dt = ∫_a^b 1. f(t) dt = b f(b) -a f(a) - ∫₀^a t f’(t) dt

              ∫_a^b f(t) dt  - b f(b) + a f(a) + ∫₀^a t f’(t) dt = 0  … (iv)

From ( iii ) , 𝜮 f(n) = k f(b) – m f(a) –  ∫_a^b [t] f’(t) dt + 0

   𝜮 f(n) = k f(b) – m f(a) –  ∫_a^b [t] f’(t) dt + ∫_a^b f(t) dt  - b f(b) + a f(a) + ∫₀^a t f’(t) dt  … ( v )

                                                                                           | since replace o by L.H.S. of (iv) |

Now 𝜮 f(n) = ∫_a^b f(t)dt + ∫_a^b (t – [t] ) dt +f(b)( k – b) -f() ( m -a )

         𝜮 f(n) = ∫_a^b f(t)dt + ∫_a^b (t – [t] ) dt +f(b)( [b] – b) -f(a) ( [a] -a ).

                                     *** Hence the Proof ***

 * big " Oh " notation 

* Bracket Function 

 * Selberg Identity

* Perfect Number

 *Historical background of Number Theory

 * Dirichlet Multiplication

 * Euclidean Algorithm

 * Fundamental Theorem of Arithmetic

 *  Properties of Numbers

 * Historical Introduction to Number Theory 

 * GCD of morethan 2 numbers

 *   The Mobius Function 𝝻 ( n ) .

 *  The Euler Totient Function 

 * Formal Power Series 

 * Liouville’s function λ(n) 

  * Congruences

 * Fermat's Theorem

 * Wilson's Theorem  

#Euler's #summation #formula #: #Number #Theory