I have a problem calculating the following perpetuity:
For given $m \in \mathbb{N}\backslash \{0\}$ we have a perpetuity, which pays out $\dfrac{j+1}{m}$, $j \in \mathbb{N}$, at the time points $$\left\{ j + \frac{k}{m}: k=0, \ldots, m-1\right\}$$ What is the cash equivalent of this perpetuity depending on the interest rate $i$?
So I know that for a perpetuity with $m$ payments of $\frac{1}{m}$, the cash equivalent is calculated via:
$$\ddot{a}_{\infty}^{(m)} = \frac{1}{m} + \frac{1}{m}v^{\frac{1}{m}} + \frac{1}{m}v^{\frac{2}{m}} + \cdots = \frac{1}{m} \sum_{k=0}^{\infty} v^{\frac{k}{m}} = \frac{1}{m(1-v^{\frac{1}{m}})} = \frac{1}{d_m}$$
where $v = \dfrac{1}{r} = \dfrac{1}{1+i}$ and $d_m = m(1-v^{\frac{1}{m}})$. So my idea was to kind of use the similar formula.
The payments are $\dfrac{j+1}{m}$, so
$$\ddot{a}_{\infty}^{(m)} = \frac{j+1}{m} \left(1+v^j + v^{j + \frac{1}{m}} + \cdots\right)= \cdots$$ but I don't actually know, if this is the right way to proceed.
The cash flow has present value $$\frac{1}{m}(1 + v^{1/m} + \cdots + v^{(m-1)/m}) + \frac{2}{m}(v + v^{1+1/m} + \cdots + v^{1+(m-1)/m}) + \cdots.$$ This of course can be written as $$\left(\frac{1}{m} + \frac{2}{m} v + \frac{3}{m}v^2 + \cdots \right) (1 + v^{1/m} + \cdots + v^{(m-1)/m}).$$ In actuarial notation, the present value is $$\frac{1}{m} (I\ddot a)_{\overline{\infty}\rceil} \ddot a_{\overline{m}\rceil}^{(m)} = \frac{1}{m(1-v)^2} \cdot \frac{1-v}{1-v^{1/m}} = \frac{1}{m(1-v)(1-v^{1/m})}.$$