Understanding the Greatest Integer Function

The greatest integer function, denoted as [x], returns the largest integer less than or equal to x. For example, [2.3] = 2, [−1.7] = −2, and [−0.5] = −1.

Step 1: Analyze the Series

The series is: $[-\frac{1}{3}]+[-\frac{1}{3}-\frac{1}{100}]+[-\frac{1}{3}-\frac{2}{100}]+\dots +[-\frac{1}{3}-\frac{99}{100}]$

Let the general term be: $a_k=[-\frac{1}{3}-\frac{k}{100}]$ for k = 0 to 99.

Note that $-\frac{1}{3}\approx -0.3333$ and $\frac{k}{100}$ ranges from 0 to 0.99.

Step 2: Determine the Value of the Greatest Integer for Each Term

We need to find the integer value of $-\frac{1}{3}-\frac{k}{100}$ for each k.

Let $x=-\frac{1}{3}-\frac{k}{100}$ . Since $-\frac{1}{3}\approx -0.3333$ , x ranges from approximately −0.3333 to −1.3233 as k goes from 0 to 99.

The greatest integer [x] will be:

• −1 when x is in (−1, 0)
• −2 when x is in (−2, −1]

Step 3: Find the Critical Point Where [x] Changes from −1 to −2

We need to find the smallest k for which $-\frac{1}{3}-\frac{k}{100}\le -1$ .

Solve the inequality:

So, for k ≥ 67, [x] = −2. For k = 0 to 66, [x] = −1.

Step 4: Count the Number of Terms

Total terms: 100 (from k=0 to k=99).

Number of terms with [x] = −1: k = 0 to 66 → 67 terms.

Number of terms with [x] = −2: k = 67 to 99 → 33 terms.

Step 5: Compute the Sum

Sum = (Number of terms with value −1) × (−1) + (Number of terms with value −2) × (−2)

Final Answer

The sum of the series is −133.

Related Topics

• Greatest Integer Function (Floor Function)
• Inequalities
• Series and Summation
• Piecewise Functions

Key Formulae and Theory

Greatest Integer Function: For any real number x, [x] is the greatest integer less than or equal to x.

Properties:

• [x] = n if n ≤ x < n+1, where n is an integer.
• For negative numbers, e.g., [−1.3] = −2, [−0.5] = −1.

Sum of a Series: Sum = Σ (value of each term). When terms have only a few distinct values, group them by value.