Question

Prove the following using the following definition of O,Big-omega,Theta, small omega

Σ^k_i=1 ?_i ?ⁱ = ?(n^k )??? ? > 1.

Answer 1

`Hey,

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Let f(n)=a(1)*n+a(2)*n^2+.....a(k)*n^k

Since we know n^k>=n^c for any c<=k

So,

f(n)=a(1)*n+a(2)*n^2+.....a(k)*n^k<=(a(1)+a(2)+a(3)+.....a(k))*n^k

Let c=(a(1)+a(2)+a(3)+.....a(k))

So,

f(n)<=c*n^k

So, by deifinition of Big O

f(n)=O(n^k)

Kindly revert for any queries

Thanks.