Why $\sum_{k=1}^\infty \left\lfloor\frac{n}{p^k}\right\rfloor≤\frac{n}{p-1}$?

Question

I need some help: can someone tell me why

$$\sum_{k=1}^\infty \left\lfloor\frac{n}{p^k}\right\rfloor≤\frac{n}{p-1}$$

I found this inequality in Wikipedia, and I want to know if it's true, thanks!

We have $\lfloor x\rfloor \leqslant x$. Then you need to know how to sum a geometric series. — Daniel Fischer, Feb 18 '15 at 14:15

score 3 · Accepted Answer · answered Feb 18 '15 at 14:16

3

$$\left\lfloor\frac{n}{p^k}\right\rfloor\leq \frac{n}{p^k} $$ and: $$\sum_{k\geq 1}\frac{1}{p^k}=\frac{1}{p-1}.$$

answered Feb 18 '15 at 14:16

Jack D'Aurizio

1 Answers1