Infinite Sequences and Series · Medium problem

The first three terms of the Maclaurin series for $\ln(1+x)$ are used to estimate $\ln\!\left(1+\dfrac1k\right)$ for a positive integer $k$, giving $\dfrac1k-\dfrac1{2k^2}+\dfrac1{3k^3}$. If this estimate equals exactly $\dfrac{137}{750}$, find $k$.