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.
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