Evaluate the sum ${1 - \frac{1}{2} {n \choose 1} + \frac{1}{3} {n \choose 2} + \ldots + (-1)^n \frac{1}{n+1} {n \choose n}}$ Evaluate the sum $${1 - \frac{1}{2} {n \choose 1} + \frac{1}{3} {n \choose 2} + \ldots + (-1)^n \frac{1}{n+1} {n \choose n}}.$$
I have tried comparing this to the similar problem here.
I believe I need to differentiate or integrate? But I'm not sure how that might work.
Any ideas? Thanks.

Hint
$${1 - \frac{1}{2} {n \choose 1} + \frac{1}{3} {n \choose 2} + \ldots + (-1)^n \frac{1}{n+1} {n \choose n}}.=\sum_{j=0}^{n} \frac{(-1)^j}{j+1} \binom n j $$
$$\sum_{j=0}^{n} \frac{(-1)^j}{j+1} x^{j+1} \binom n j =\int \sum_{j=0}^{n} (-1)^j x^{j}\binom n j =\int (1-x)^n=- \frac { (1-x)^{n+1}} {n+1} $$
$$\sum_{j=0}^{n} \frac{(-1)^j}{j+1} \binom n j =\int_0^1 (1-x)^n=\frac 1 {n+1}$$
