Logarithmic series from an integral Taylor remainder ($-\log (1-t)={\sum }_{k=1}^{\infty }{t}^k/k,0\le t<1$)

The integral remainder in Taylor theorem for $\log (1-t)$ gives the exact finite identityFor $0\le x\le t<1$, $(t-x)/(1-x)\le t$, so the nonnegative remainder is at most $t^{N+1}/(1-t)$ and tends to zero. At $t=1/2$ it is at most $2^{-N}$, proving both convergence to $\log 2$ and an explicit error bound without assuming the logarithmic power series beforehand.