The question : At a nominal rate of interest $i$, convertible semiannually, the present value of a series of payments of $1$ at the end of every $2$ years, forever, is $5.89$. Calculate $i$.
I attempted this by solving for the PV of a perpetuity $5.89 = \frac{1}{i}$ which is $i=0.16977$
Then I plug it into the formula $1+i=(1+\frac{i^{(2)}}{2})^2$ and $i^{(2)}= 0.163188$
I know the final answer is $i=8\%$, so where did I make a mistake?
EDIT: The problem was that I was raising my expression to the second power instead of the fourth. I overlooked that it was being compounded every $2$ years. So semi-annual for two years equals $4$ periods, therefore I need to raise it to the fourth power. $1+i=(1+\frac{i^{(2)}}{2})^4$ and $i^{(2)}= 0.07996$
The effective bi-annual interest rate is $j$ such that $$ \left(1+\frac{i^{(2)}}{2}\right)^4=1+j $$ So you have $$ 5.89=a_{\overline{\infty}|j}=\frac{1}{j}\quad\Longrightarrow\quad j=\frac{1}{5.89} $$ and $$ i^{(2)}=2\left[\left(1+j\right)^{1/4}-1\right]\approx 7.99648\% $$ so we can say that the interest converttible semiannually is $i=i^{(2)}\approx 8\%$.