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Evaluate: $C_0+\frac{C_1}2+\frac{C_2}3+\cdots\frac{C_n}{n+1}$.Where $C_k$ denote the usual binomial coefficient.

MY TRY: I solved it and got $\frac{2^{n+1}}{n+1}$,but it's ans. is given as $\frac{2^{n+1}-1}{n+1}$.Thank you.

2 Answers 2

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HINT:

$$(1+x)^n=\sum_{r=0}^n\binom nr x^r$$

$$\int_0^1(1+x)^n=\sum_{r=0}^n\binom nr\int_0^1x^r$$

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    @MatheMagic, I would like to know your method, though! – 2017-01-18
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    Same as you did by integration but i forgot to put zero in LHS after integration.That's why i skipped 1. – 2017-01-18
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    @MatheMagic, Can you please try http://math.stackexchange.com/questions/437523/proving-binomial-idenity-without-calculus without the help of calculus – 2017-01-18
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$$\displaystyle\dfrac{\binom nr}{r+1}=\cdots=\dfrac{\binom{n+1}{r+1}}{n+1}$$

$$\displaystyle\implies\sum_{r=0}^n\dfrac{\binom nr}{r+1}=\dfrac1{n+1}\sum_{r=0}^n\binom{n+1}{r+1}$$ Now, $$\displaystyle\sum_{r=0}^n\binom{n+1}{r+1}=-\binom{n+1}0+\sum_{r=-1}^n\binom{n+1}{r+1}=-1+(1+1)^{n+1}$$