$$\sum_{r\ =\ 1}^{\infty}\frac{3}{r\left(r\ +\ 1\right)}=$$

The sum in question is:

\[
\sum_{r=1}^{\infty} \frac{3}{r(r+1)}.
\]

### Step-by-Step Solution

1. **Partial Fraction Decomposition**: 

   Let's break down the expression $ \frac{3}{r(r+1)} $ using partial fractions. We seek constants $ A $ and $ B $ such that:

   \[
   \frac{3}{r(r+1)} = \frac{A}{r} + \frac{B}{r+1}.
   \]

   Multiplying through by $ r(r+1) $, we get:

   \[
   3 = A(r + 1) + Br.
   \]

   Simplifying this, we have:

   \[
   3 = (A + B)r + A.
   \]

   For this equation to hold for all $ r $, the coefficients of $ r $ and the constant term must match on both sides. Therefore, we obtain the system of equations:

   \[
   A + B = 0, \quad A = 3.
   \]

   Solving this, we get $ A = 3 $ and $ B = -3 $. So, the partial fraction decomposition is:

   \[
   \frac{3}{r(r+1)} = \frac{3}{r} - \frac{3}{r+1}.
   \]

2. **Rewrite the Series**:

   Substitute the partial fractions into the sum:

   \[
   \sum_{r=1}^{\infty} \frac{3}{r(r+1)} = \sum_{r=1}^{\infty} \left( \frac{3}{r} - \frac{3}{r+1} \right).
   \]

3. **Telescoping Series**:

   This is a telescoping series, where most terms cancel out. Let's write out the first few terms to see the pattern:

   \[
   \left(\frac{3}{1} - \frac{3}{2}\right) + \left(\frac{3}{2} - \frac{3}{3}\right) + \left(\frac{3}{3} - \frac{3}{4}\right) + \dots.
   \]

   Notice that almost every positive term $ \frac{3}{r} $ cancels with the subsequent negative term $ -\frac{3}{r} $ in the next fraction. Therefore, after cancellation, we are left with:

   \[
   3 - \lim_{n \to \infty} \frac{3}{n+1}.
   \]

4. **Evaluate the Limit**:

   Since $ \lim_{n \to \infty} \frac{3}{n+1} = 0 $, the remaining sum is:

   \[
   3.
   \]

### Final Answer

\[
\sum_{r=1}^{\infty} \frac{3}{r(r+1)} = 3.
\]

Would you like more details or have any questions?

### Related Questions

1. How would you find the sum

$$\sum_{r\ =\ 1}^{\infty}\frac{3}{r\left(r\ +\ 1\right)}$$

Explore the detailed step-by-step solution to sum the infinite series $$\sum_{r=1}^{\infty}\frac{3}{r(r+1)}$$ using partial fraction decomposition and telescoping series techniques. Learn how to evaluate the limit to get the final sum. Suitable for advanced high school and early college students.

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