prove that sum(1/(x(x+1))) from 1 to infinity is 1

We are tasked with proving that the infinite sum \( \sum_{x=1}^{\infty} \frac{1}{x(x+1)} = 1 \).

### Step-by-Step Solution

#### Step 1: Decompose the general term
We start by simplifying the general term \( \frac{1}{x(x+1)} \). This can be decomposed using **partial fraction decomposition**:

\[
\frac{1}{x(x+1)} = \frac{A}{x} + \frac{B}{x+1}
\]

Multiplying both sides by \( x(x+1) \), we get:

\[
1 = A(x+1) + Bx
\]

Simplifying:

\[
1 = A(x+1) + Bx = Ax + A + Bx = (A+B)x + A
\]

For this equation to hold for all \( x \), we must have:
- \( A + B = 0 \)
- \( A = 1 \)

From \( A = 1 \), it follows that \( B = -1 \).

Thus, the decomposition is:

\[
\frac{1}{x(x+1)} = \frac{1}{x} - \frac{1}{x+1}
\]

#### Step 2: Write the sum using the decomposition
Now we rewrite the infinite sum:

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

This is a **telescoping series**, where most terms cancel out.

#### Step 3: Simplify the telescoping series
Writing out the first few terms:

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

Notice that all terms cancel out except for the first term \( \frac{1}{1} \), leaving:

\[
1
\]

#### Step 4: Conclusion
Thus, the sum of the series is:

\[
\sum_{x=1}^{\infty} \frac{1}{x(x+1)} = 1
\]

This completes the proof.

---
Would you like further clarification or additional details? Here are 5 related questions:

1. How do telescoping series work in more complicated sums?
2. Can partial fraction decomposition be used for other types of series?
3. What are other examples of series that converge to simple values?
4. How can this technique be applied to sums that don't start at 1?
5. What other methods exist for proving the convergence of an infinite series?

**Tip**: Always look for patterns like telescoping behavior when summing series—it often simplifies the problem greatly!

Prove that sum(1/(x(x+1))) from 1 to infinity is 1.

Learn how to prove that the infinite sum of 1/(x(x+1)) from 1 to infinity equals 1 using partial fraction decomposition and the telescoping series method. This problem demonstrates advanced series techniques suitable for students in Grades 11-12.

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