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Mathematical Analysis Fundamentals by Agamirza Bashirov

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Proof

Equality (a) is exactly the binomial formula:

k=0nn!k!(n-k)!xk(1-x)n-k=(x+1-x)n=1.

image

Equality (b) can be reduced to the binomial formula as well:

k=0nn!k!(n-k)!xk(1-x)n-kk=nxk=1n(n-1)!(k-1)!(n-k)!xk-1(1-x)n-k=nxk=0n-1(n-1)!k!(n-k-1)!xk(1-x)n-k-1=nx(x+1-x)n-1=nx.

image

In a similar way, using k2=k+k(k-1)image, we obtain

k=0nn!k!(n-k)!xk(1-x)n-kk2=k=0nn!k!(n-k)!xk(1-x)n-kk+k=0nn!k!(n-k)!xk(1-x)n-kk(k-1).

Here the first term is equal to nx by ...

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