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

Question

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 $\frac{n}{3n-1}.$

2. 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 $x+\frac{x^5}{5}+\frac{x^7}{7}+\cdots.$

[Note that the original question had $\log$ meaning ‘$\log$ to base $e$’.]