Alternating Harmonic Series of Binomial Coefficients

If $c_0,c_1,c_2,\dots ,c_n$ are the coefficients in the expansion of $(1+x{)}^n$, where $n$ is a positive integer, show that $c_1-\frac{c_2}{2}+\frac{c_3}{3}-\dots +\frac{(-1{)}^{n-1}{c}_n}{n}=1+\frac{1}{2}+\frac{1}{3}+\dots +\frac{1}{n}$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XIV. Binomial Theorem: Any Index, Examples XIV.c, No. 23