How can one show that [math]\displaystyle \sum_{k=1}^n (-1)^{k-1} k^2 = (-1)^{n-1} \, \frac{n (n+1)}{2}[/math] for [math]n \geq 1[/math]?
How can one show that [math]\sum_{k=1}^n\frac{(-1)^{k+1}\binom n k}k = \sum_{k=1}^n \frac 1 k[/math]?