Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 19 / a / Solution

For $r\ne 1$, differentiate the finite geometric series and multiply by $r$:The required terms are $-n{r}^n$ with $r=-1/3$. The finite weighted geometric sum therefore givesSince $N/3^N\to 0$, the correction tends to zero, proving convergence and yieldingIn fact the ratio test also proves absolute convergence, since the ratio of consecutive absolute terms tends to $1/3$.