Review question

# Can we show the sum of this series to $n$ terms is $n/(3n-1)$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5634

## Suggestion

1. Show that the sum to $n$ terms of the series $\frac{1}{2.1}-\frac{1}{5.2}-\frac{1}{8.5}-\cdots+ \frac{1}{(3r-1)(4-3r)}\cdots$ is $\frac{n}{3n-1}.$

Could we try a proof by induction here?

Or else, could we split $\dfrac{1}{(3r-1)(4-3r)}$ into partial fractions?

1. Given that for $-1<x<+1, \quad \log_e(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}- \quad\text{to infinity,}$ deduce the sum to infinity of $x+\frac{x^5}{5}+\frac{x^7}{7}+\cdots.$

How does the sequence continue? Could it be $x+\frac{x^5}{5}+\frac{x^7}{7}+\frac{x^9}{9}+\frac{x^{11}}{11}+\cdots,$ so that the only odd power missing here is $\dfrac{x^3}{3}$?
What would the series for $\log_e(1-x)$ be? Or the series for $\log_e(1+x^2)$?