How to formally write $f\left(k\right)={1\over p_1}\left(1+{1\over p_2}\left(1+{1\over p_3}\left(1+\dots\right)\right)\right)$

Question

How do I write the following finite series as a sum or product:

$$f\left(k\right) = {1 \over p_1} \left(1 + {1 \over p_2} \left(1 + {1 \over p_3}\left(1+\dots \right) \right) \right)$$

…all the way up to $p_k$, the $k$th prime? $p_1 = 2$.

Also, how can I actually solve it for a given (large) $k$, without loosing precision due to floating point arithmetic?

Answer 1

That's: $f(k) = \dfrac 1 {p_1} + \dfrac 1 {p_1p_2}+ \dfrac 1 {p_1p_2p_3} + \dotsc + \dfrac 1 {p_1 p_2 p_3 \cdots p_k}= \sum\limits_{n=1}^k \left(\prod\limits_{m=1}^n \dfrac 1 {p_m}\right)$