# Another transposed harmonic alternating series

**Calculus**Level 5

\[\large \sum_{k=1}^{c} \left(\dfrac{1}{2k}\right)-1+\sum_{k=c+1}^{2c} \left(\dfrac{1}{2k}\right)-\dfrac{1}{3}+\sum_{k=2c+1}^{3c}\left(\dfrac{1}{2k}\right)-\dfrac{1}{5}+\cdots=\dfrac{1}{2}\ln\frac{5}{4}\]

In the previous series, the number \(c\) is a positive integer and the sum of the series is the number \(\dfrac{1}{2}\ln\dfrac{5}{4}\). Find the value of \(c\) if there is any, otherwise, enter a 666 as your answer.