Prove that `2^k(n,0)(n,k)-2^(k-1)(n,1)(n-1,k-1)+2^(k-2)(n,2)(n-2,k-2)-.....+(-1)^k(n,k)(n-k,0)=(n,k)`.

If n and k are positive integers, show that 2^k(nC 0)(n k)-2^(k-1)(nC1)(n-1Ck-1)+2^(k-2)(nC2)((n-2k-2))_dot-...+(-1)^k(nCk)+(n-kC0)=(nC k)w h e r e(n C k) stands for ^n C_kdot

If n and k are positive integers, show that 2^k( .^n C_0)(.^n C_k)-2^(k-1)(.^n C_1)(.^(n-1) C_k-1)+2^(k-2)(.^n C_2)((n-2k-2))_dot-...+ (-1)^k(^n C_k)+(.^(n-k) C_0)=(.^n C_k)w h e r e(.^n C_k) stands for .^n C_k.

2 ^ (k) ([n0]) ([nk]) - 2 ^ (k-1) ([n1]) ([n-1k-1]) + 2 ^ (k-2) ([n2]) ([nk-2]) - ,,, + (- 1) ^ (k) ([nk]) ([nk]) ([n0]) ([n0])

prove that sum_(k=1)^(n)k2^(-k)=2[1-2^(-n)-n*2^(-(n+1)))

If Delta_k=|(1,n,n),(2(k-1),n(n+1),n(n-1)),(4k(k^2-1),n(n+1)(n^2-4),n(n+2)(n^2-1))|, then f(n)=sum_(k=1)^n Delta_k+n is a polynomial of degree (i)0 (ii)1 (iii)6 (iv)7

{:(" "Lt),(n rarr oo):} (2^(k)+4^(k)+6^(k)+....+(2n)^(k))/(n^(k+1))=

lim_(n toinfty)[(1^(k) +2^(k)+3^(k) +. . .. +n^(k))/(n^(k +1))]=

For the natural numbers n,k,<=tT(n,k)=(1)/((n+1)!)+(1)/((n+2)!)+(1)/((n+3)!)+......(1)/((n+k)!)