It's a problem from a high school math book that I've been unable to solve:
Prove using definite integrals that, $${n \choose 1}-\frac{1}{2}{n \choose 2}+\frac{1}{3}{n \choose 3}-...+(-1)^{n-1}\frac{1}{n}{n \choose n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$$
I've tried using binomial expansions but to no avail. I cannot see any other approach to it using definite integrals. Although I am not sure if this is true. So, if one cannot prove it, can you please at least give me some numerical evidence of its correctness. A proof will be much better though.
The right-hand side is $$\int_0^11+x+x^2+...+x^{n-1}dx=\int_0^1\frac{1-x^n}{1-x}dx$$ The left-hand side is a similar integral, with $x$ replaced by $1-x$.