For xR, let [x] denote the greatest integer x.

Question

For $x\in\mathbb{R}$, let $[x]$ denote the greatest integer $\le x$. Then the sum of the series
$$\left[-\frac13\right]+\left[-\frac13-\frac{1}{100}\right]+\left[-\frac13-\frac{2}{100}\right]+\cdots+\left[-\frac13-\frac{99}{100}\right]$$
is:

Accepted Answer

Using $[x]+[-x]=-1$ (for non-integer $x$) and the identity $\left[x+\tfrac1n\right]+\left[x+\tfrac2n\right]+\cdots = [nx]-\ldots$:

$$\sum_{k=0}^{99}\left[-\frac13-\frac{k}{100}\right] = -100-\sum_{k=0}^{99}\left(\frac13+\frac{k}{100}-\left[\ldots\right]\right)\cdots = -100-\frac{100}{3} = -133.$$

Solutions · 1