[Math] How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$


Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$

Apart from induction, I tried with Wolfram Alpha to check the validity, but I can't think of an easy (manual) alternative.

Please suggest an intuitive/easy method.

Best Answer

Look at $$\int_0^1(1-x)^n dx$$ This is easy to compute by substitution.

Now compute it the hard way, by expanding using the Binomial Theorem, and integrating term by term.

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