algebra precalculus - How to prove $sumlimits_{r=0}^n frac{(-1)^r}{r+1}binom{n}{r} = frac1{n+1}$?

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.

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.