Alg10.20

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Prove that {C \choose 0}+\frac{{C \choose 1}}{2} x+\frac{{C \choose 2}}{3} x^2+.....+\frac{{C \choose n}}{n+1} x^n=\frac{(1+x)^{n+1}-1}{(n+1)x}\,

{C \choose 0}+\frac{{C \choose 1}}{2} x+\frac{{C \choose 2}}{3} x^2+.....+\frac{{C \choose n}C_n}{n+1} x^n=1+\frac{n}{a!}\cdot\frac{x}{2}+\frac{n(n-1)}{2}\cdot\frac{x^2}{3}+........\,

1+\frac{n}{2!} x+\frac{n(n-1)}{3!} x^2+.........\,

\frac{1}{(n+1)x}[\frac{(n+1)x}{1!}+\frac{(n+1)n) x^2}{2!}+\frac{(n+1)n(n-1) x^3}{3!}+......]\,

\frac{1}{(n+1)x}[(n+1)\!C_1 \cdot x+(n+1)\!C_2 \cdot x^2+(n+1)\!C_3\cdot x^3+.......]\,

\frac{1}{(n+1)x}[(n+1)\!C_1 \cdot x+(n+1)\!C_2 \cdot x^2+(n+1)\!C_3\cdot x^3+.......+(n+1)\!C_{n-1}\cdot x^{n+1}-1]\,

\frac{1}{(n+1)x}[(1+x)^{n+1}-1]=\frac{(1+x)^{n+1}-1}{(n+1)x}\,


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