Recognize $4-\frac{4^{2}}{2}+\frac{4^{3}}{3}-\frac{4^{4}}{4}+\cdots$ As a Taylor Series Evaluated at a Particular Value of Value

Recognize $4-\frac{4^{2}}{2}+\frac{4^{3}}{3}-\frac{4^{4}}{4}+\cdots$ as a Taylor series evaluated at a particular value of x and find the sum to 4 decimal places.