C 0- C 1 x -2- C 2 X 2-3- C n X n - n -1-A- 1-n-1 xB- 1-xn-n-1 xC- 1-xn-1-n-1 xD- 1-xn-1-1-n-1 x

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Integration to solve modified sum of binomial coefficients

C 0+ C 1 x /2...

Question

$C_{0} + \frac{C_{1} x}{2} + \frac{C_{2} X^{2}}{3} + \dots \frac{C_{n} X^{n}}{n + 1}$ =

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Solution

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(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . C_{n} x^{n}

3 C_{0} + 3^{2} \frac{C_{1}}{2}

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(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . + C_{n} x^{n}

\frac{n!}{x (x + 1) (x + 2) (x + 3) . . . (x + n)} = \frac{C_{0}}{x} - \frac{C_{1}}{x + 1} - . . . + (- 1)^{n} \frac{C_{n}}{x + n}

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If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . . + C_{n} x^{n}$ , then

$\frac{C_{1}}{C_{0}}$ + $\frac{2 C_{2}}{C_{1}}$ + $\frac{3 C_{3}}{C_{2}}$ + ........+ $\frac{n C_{n}}{C_{n - 1}}$ =

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + C_{3} x^{3} + \dots + C_{n} x^{n},

C_{0} C_{2} + C_{1} C_{3} + C_{2} C_{4} + \dots + C_{n - 2} C_{n} =

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