Infinite Sequences and Series · Medium problem

A feedback controller's total correction over all rounds is modeled by the infinite series $\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}\dfrac{(1/k)^n}{n}$, for a positive integer $k$ (this is the Maclaurin series for $\ln(1+x)$ at $x=\dfrac1k$). If the total correction equals exactly $\ln\!\left(\dfrac54\right)$, find $k$.