\displaystyle\lim_{n \to \infty}\sum_{k=1}^n(-1)^{k-1}\cdot4^{-k+2} &=& \sum_{n=1}^\infty(-1)^{n-1}\cdot4^{-n+2} = \\ = 4\sum_{n=1}^\infty(-1)^{n-1}\cdot4^{-n+1} = 4\sum_{n=0}^\infty(-1)^n\cdot4^{-n} =4\sum_{n=0}^\infty\left(-\frac{1}{4}\right)^n =\\ = 4\cdot\frac{1}{1-\left(-\frac{1}{4}\right)} = \frac{16}{5}