Prove $\binom{n}{1} - \frac{1}{2}\binom{n}{2} + \frac{1}{3}\binom{n}{3} + \ldots = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}$

Question

I thought of induction but that was unwieldy. Then considered using $(1+x)^n$. Didn't succeed.

Any hint? Thanks