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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.

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    Is this just a matter of convention? I guess I'm not clear on what else sums with non-integer upper bounds could even mean.2011-12-05

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