What is the value of [math] \displaystyle \sum_{n=1}^{\infty }{\frac{{{\left( -1 \right)}^{n+1}}}{4{{n}^{2}}-1}} [/math] ?
If [math]n\in \mathbb{N}[/math], what is the value of [math]\sum_{k=1}^{n}k^3\cdot\left(\frac{^{n}C_{k}}{^{n}C_{k-1}}\right) ^3 [/math]?