Monday 15 September 2014

Summation and series

Successive difference formulae for the series having constant kth successive difference

f(n)=sum_(k=0)^(p)alpha_k(n; k)

=a_0+b_0n+(c_0n(n-1))/2+(d_0n(n-1)(n-2))/(2·3)+....
            1 19 143 607 1789 4211 8539 18 124 464 1182 2422 4328 106 340 718 1240 1906 234 378 522 666 144 144 144 0 0
           
f(n)=1+18n+1/2106n(n-1)+1/6234n(n-1)(n-2)+1/(24)144n(n-1)(n-2)(n-3)
                    
 =6n^4+3n^3+2n^2+7n+1,


 SUMMATION FORMULAE
   
sum_(k=1)^(n)k=1/2(n^2+n)

sum_(k=1)^(n)k^2=1/6(2n^3+3n^2+n)

sum_(k=1)^(n)k^3=1/4(n^4+2n^3+n^2)

sum_(k=1)^(n)k^4=1/(30)(6n^5+15n^4+10n^3-n)
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