Can anyone tell me how to approximate the following functions?
$f_3(n) = \displaystyle\sum_{j=1}^n\sum_{k=1}^{\lfloor\frac{n}{j}\rfloor}\int_1^{\frac{n}{jk}}dx$
$f_4(n) = \displaystyle\sum_{j=1}^n\sum_{k=1}^{\lfloor\frac{n}{j}\rfloor}\sum_{l=1}^{\lfloor\frac{n}{jk}\rfloor}\int_1^{\frac{n}{jkl}}dx$
$f_5(n) = \displaystyle\sum_{j=1}^n\sum_{k=1}^{\lfloor\frac{n}{j}\rfloor}\sum_{l=1}^{\lfloor\frac{n}{jk}\rfloor}\sum_{m=1}^{\lfloor\frac{n}{jkl}\rfloor}\int_1^{\frac{n}{jklm}}dx$
and so on for $f_k(n)$ in general - and especially how to tell what the error term would be in such approximations?
I know from graphing them that they all look like quite smooth functions.